Copyright2008‐13,EarlWhitney,RenoNV.AllRightsReserved

MathHandbook

ofFormulas,ProcessesandTricks

Algebra

Preparedby:EarlL.Whitney,FSA,MAAA

Version2.5

April2,2013

Page Description

Chapter1:Basics9 OrderofOperations(PEMDAS,ParentheticalDevice)10 GraphingwithCoordinates(Coordinates,PlottingPoints)11 LinearPatterns(Recognition,ConvertingtoanEquation)12 IdentifyingNumberPatterns13 CompletingNumberPatterns14 BasicNumberSets(SetsofNumbers,BasicNumberSetTree)

Chapter2:Operations15 OperatingwithRealNumbers(AbsoluteValue,Add,Subtract,Multiply,Divide)16 PropertiesofAlgebra(Addition&Multiplication,Zero,Equality)

Chapter3:SolvingEquations18 SolvingMulti‐StepEquations19 TipsandTricksinSolvingMulti‐StepEquations

Chapter4:Probability&Statistics20 ProbabilityandOdds21 ProbabilitywithDice22 Combinations23 StatisticalMeasures

Chapter5:Functions24 IntroductiontoFunctions(Definitions,LineTests)25 SpecialIntegerFunctions26 OperationswithFunctions27 CompositionofFunctions28 InversesofFunctions29 Transformation�Translation30 Transformation�VerticalStretchandCompression31 Transformation�HorizontalStretchandCompression32 Transformation�Reflection33 Transformation�Summary34 BuildingaGraphwithTransformations

AlgebraHandbookTableofContents

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AlgebraHandbookTableofContents

Page Description

Chapter6:LinearFunctions35 SlopeofaLine(MathematicalDefinition)36 SlopeofaLine(RiseoverRun)37 SlopesofVariousLines(8Variations)38 VariousFormsofaLine(Standard,Slope‐Intercept,Point‐Slope)39 SlopesofParallelandPerpendicularLines40 Parallel,PerpendicularorNeither41 Parallel,CoincidentorIntersecting

Chapter7:Inequalities42 PropertiesofInequality43 GraphsofInequalitiesinOneDimension44 CompoundInequalitiesinOneDimension45 InequalitiesinTwoDimensions46 GraphsofInequalitiesinTwoDimensions47 AbsoluteValueFunctions(Equations)48 AbsoluteValueFunctions(Inequalities)

Chapter8:SystemsofEquations49 GraphingaSolution50 SubstitutionMethod51 EliminationMethod52 ClassificationofSystemsofEquations53 LinearDependence54 SystemsofInequalitiesinTwoDimensions55 ParametricEquations

Chapter9:Exponents(Basic)andScientificNotation56 ExponentFormulas57 ScientificNotation(Format,Conversion)58 AddingandSubtractingwithScientificNotation59 MultiplyingandDividingwithScientificNotation

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AlgebraHandbookTableofContents

Page Description

Chapter10:Polynomials�Basic60 IntroductiontoPolynomials61 AddingandSubtractingPolynomials62 MultiplyingBinomials(FOIL,Box,NumericalMethods)63 MultiplyingPolynomials64 DividingPolynomials65 FactoringPolynomials66 SpecialFormsofQuadraticFunctions(PerfectSquares)67 SpecialFormsofQuadraticFunctions(DifferencesofSquares)68 FactoringTrinomials�SimpleCaseMethod69 FactoringTrinomials�ACMethod70 FactoringTrinomials�BruteForceMethod71 FactoringTrinomials�QuadraticFormulaMethod72 SolvingEquationsbyFactoring

Chapter11:QuadraticFunctions73 IntroductiontoQuadraticFunctions74 CompletingtheSquare75 TableofPowersandRoots76 TheQuadraticFormula77 QuadraticInequalitiesinOneVariable79 FittingaQuadraticthroughThreePoints

Chapter12:ComplexNumbers80 ComplexNumbers‐Introduction81 OperationswithComplexNumbers82 TheSquareRootofi83 ComplexNumbers�GraphicalRepresentation84 ComplexNumberOperationsinPolarCoordinates85 ComplexSolutionstoQuadraticEquations

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AlgebraHandbookTableofContents

Page Description

Chapter13:Radicals86 RadicalRules87 SimplifyingSquareRoots(ExtractingSquares,ExtractingPrimes)88 SolvingRadicalEquations89 SolvingRadicalEquations(PositiveRoots,TheMissingStep)

Chapter14:Matrices90 AdditionandScalarMultiplication91 MultiplyingMatrices92 MatrixDivisionandIdentityMatrices93 Inverseofa2x2Matrix94 CalculatingInverses�TheGeneralCase(Gauss‐JordanElimination)95 Determinants�TheGeneralCase96 Cramer’sRule�2Equations97 Cramer’sRule�3Equations98 AugmentedMatrices99 2x2AugmentedMatrixExamples100 3x3AugmentedMatrixExample

Chapter15:ExponentsandLogarithms101 ExponentFormulas102 LogarithmFormulas103 e104 TableofExponentsandLogs105 ConvertingBetweenExponentialandLogarithmicForms106 ExpandingLogarithmicExpressions107 CondensingLogarithmicExpressions108 CondensingLogarithmicExpressions�MoreExamples109 GraphinganExponentialFunction110 FourExponentialFunctionGraphs111 GraphingaLogarithmicFunction114 FourLogarithmicFunctionGraphs115 GraphsofVariousFunctions116 ApplicationsofExponentialFunctions(Growth,Decay,Interest)117 SolvingExponentialandLogarithmicEquations

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AlgebraHandbookTableofContents

Page Description

Chapter16:Polynomials�Intermediate118 PolynomialFunctionGraphs119 FindingExtremawithDerivatives120 FactoringHigherDegreePolynomials�SumandDifferenceofCubes121 FactoringHigherDegreePolynomials�VariableSubstitution122 FactoringHigherDegreePolynomials�SyntheticDivision123 ComparingSyntheticDivisionandLongDivision124 ZerosofPolynomials�DevelopingPossibleRoots125 ZerosofPolynomials�TestingPossibleRoots126 IntersectionsofCurves(GeneralCase,TwoLines)127 IntersectionsofCurves(aLineandaParabola)128 IntersectionsofCurves(aCircleandanEllipse)

Chapter17:RationalFunctions129 DomainsofRationalFunctions130 HolesandAsymptotes131 GraphingRationalFunctions131 SimpleRationalFunctions132 SimpleRationalFunctions‐Example133 GeneralRationalFunctions135 GeneralRationalFunctions‐Example137 OperatingwithRationalExpressions138 SolvingRationalEquations139 SolvingRationalInequalities

Chapter18:ConicSections140 IntroductiontoConicSections141 ParabolawithVertexattheOrigin(StandardPosition)142 ParabolawithVertexatPoint(h,k)143 ParabolainPolarForm144 Circles145 EllipseCenteredontheOrigin(StandardPosition)146 EllipseCenteredatPoint(h,k)147 EllipseinPolarForm148 HyperbolaCenteredontheOrigin(StandardPosition)149 HyperbolaCenteredatPoint(h,k)150 HyperbolainPolarForm151 HyperbolaConstructionOvertheDomain:0to2152 GeneralConicEquation‐Classification153 GeneralConicFormula�Manipulation(Steps,Examples)154 ParametricEquationsofConicSections

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AlgebraHandbookTableofContents

Page Description

Chapter19:SequencesandSeries155 IntroductiontoSequencesandSeries156 FibonacciSequence157 SummationNotationandProperties158 SomeInterestingSummationFormulas159 ArithmeticSequences160 ArithmeticSeries161 PythagoreanMeans(Arithmetic,Geometric)162 PythagoreanMeans(Harmonic)163 GeometricSequences164 GeometricSeries165 AFewSpecialSeries(,e,cubes)166 Pascal’sTriangle167 BinomialExpansion168 GammaFunctionandn !169 GraphingtheGammaFunction

170 Index

UsefulWebsites

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http://mathworld.wolfram.com/

http://www.purplemath.com/

http://www.math.com/homeworkhelp/Algebra.html

WolframMathWorld�PerhapsthepremiersiteformathematicsontheWeb.Thissitecontainsdefinitions,explanationsandexamplesforelementaryandadvancedmathtopics.

PurpleMath�AgreatsitefortheAlgebrastudent,itcontainslessons,reviewsandhomeworkguidelines.Thesitealsohasananalysisofyourstudyhabits.TaketheMathStudySkillsSelf‐Evaluationtoseewhereyouneedtoimprove.

Math.com�HasalotofinformationaboutAlgebra,includingagoodsearchfunction.

Mathguy.us�DevelopedspecificallyformathstudentsfromMiddleSchooltoCollege,basedontheauthor'sextensiveexperienceinprofessionalmathematicsinabusinesssettingandinmathtutoring.Containsfreedownloadablehandbooks,PCApps,sampletests,andmore.

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AlgebraHandbookTableofContents

Schaum’sOutlines

Algebra 1 , by James Schultz, Paul Kennedy, Wade Ellis Jr, and Kathleen Hollowelly. Algebra 2 , by James Schultz, Wade Ellis Jr, Kathleen Hollowelly, and Paul Kennedy.

Although a significant effort was made to make the material in this study guide original, some material from these texts was used in the preparation of the study guide.

AnimportantstudentresourceforanyhighschoolmathstudentisaSchaum’sOutline.Eachbookinthisseriesprovidesexplanationsofthevarioustopicsinthecourseandasubstantialnumberofproblemsforthestudenttotry.Manyoftheproblemsareworkedoutinthebook,sothestudentcanseeexamplesofhowtheyshouldbesolved.

Schaum’sOutlinesareavailableatAmazon.com,Barnes&Noble,Bordersandotherbooksellers.

Note: This study guide was prepared to be a companion to most books on the subject of High School Algebra. In particular, I used the following texts to determine which subjects to include in this guide.

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AlgebraOrderofOperations

Tothenon‐mathematician,theremayappeartobemultiplewaystoevaluateanalgebraicexpression.Forexample,howwouldon llowing?eevaluatethefo

3 4 7 6 5

Youcouldworkfromlefttoright,oryoucouldworkfromrighttoleft,oryoucoulddoanynumberofotherthingstoevaluatethisexpression.Asyoumightexpect,mathematiciansdonotlikethisambiguity,sotheydevelopedasetofrulestomakesurethatanytwopeopleevaluatinganexpressionwouldgetthesameanswer.

PEMDAS

Inordertoevaluateexpressionsliketheoneabove,mathematicianshavedefinedanorderofoperationsthatmustbefollowedtogetthecorrectvaluefortheexpression.TheacronymthatcanbeusedtorememberthisorderisPEMDAS.Alternatively,youcouldusethemnemonicphrase“PleaseExcuseMyDearAuntSally”ormakeupyourownwaytomemorizetheorderofoperations.ThecomponentsofPEMDASare:

P AnythinginParenthesesisevaluatedfirst.Usuallywhentherearemultipleoperationsinthesamecategory,forexample3multiplications,theycanbeperformedinanyorder,butitiseasiesttoworkfromlefttoright.

E ItemswithExponentsareevaluatednext.

M Multiplicationand�

D Divisionareperformednext.

A Additionand�

S Subtractionareperformedlast.

ParentheticalDevice.Ausefuldeviceistouseapplyparenthesestohelpyouremember

theorderofoperationswhenyouevaluateanexpression.Parenthesesareplacedaroundtheitemshighestintheorderofoperations;thensolvingtheproblembecomesmorenatural.UsingPEMDASandthisparenthe solvetheexpressionaboveasfollows:ticaldevice,we

InitialExpression: 3 4 7 6 5

Addparentheses/brackets: 5

Note:Anyexpressionwhichisambiguous,liketheoneabove,ispoorlywritten.Studentsshouldstrivetoensurethatanyexpressionstheywriteareeasilyunderstoodbyothersandbythemselves.Useofparenthesesandbracketsisagoodwaytomakeyourworkmoreunderstandable.

3 4 7 6

SolveusingPEMDAS: 84 6 25

150 84

FinalAnswer 234

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AlgebraGraphingwithCoordinates

Graphsintwodimensionsareverycommoninalgebraandareoneofthemostcommonalgebraapplicationsinreallife.

y

CoordinatesQuadrant2 Quadrant1

Theplaneofpointsthatcanbegraphedin2dimensionsiscalledtheRectangularCoordinatePlaneortheCartesianCoordinatePlane(namedaftertheFrenchmathematicianandphilosopherRenéDescartes).

x

Quadrant3 Quadrant4

• Twoaxesaredefined(usuallycalledthex‐andy‐axes).

• Eachpointontheplanehasanxvalueandayvalue,writtenas:(xvalue,yvalue)

• Thepoint(0,0)iscalledtheorigin,andisusuallydenotedwiththeletter“O”.

• Theaxesbreaktheplaneinto4quadrants,asshownabove.TheybeginwithQuadrant1wherexandyarebothpositiveandincreasenumericallyinacounter‐clockwisefashion.

PlottingPointsonthePlane

Whenplottingpoints,

• thex‐valuedetermineshowfarright(positive)orleft(negative)oftheoriginthepointisplotted.

• They‐valuedetermineshowfarup(positive)ordown(negative)fromtheoriginthepointisplotted.

Examples:

Thefollowingpointsareplottedinthefiguretotheright:

A=(2,3) inQuadrant1B=(‐3,2) inQuadrant2C=(‐2,‐2) inQuadrant3D=(4,‐1) inQuadrant4O=(0,0) isnotinanyquadrant

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Algebra

LinearPatternsRecognizingLinearPatterns

Thefirststeptorecognizingapatternistoarrangeasetofnumbersinatable.Thetablecanbeeitherhorizontalorvertical.Here,weconsiderthepatterninahorizontalformat.Moreadvancedanalysisgenerallyusestheverticalformat.

Considerthispattern:

x‐value 0 1 2 3 4 5y‐value 6 9 12 15 18 21

Toanalyzethepattern,wecalculatedifferencesofsuccessivevaluesinthetable.Thesearecalledfirstdifferences.Ifthefirstdifferencesareconstant,wecanproceedtoconvertingthepatternintoanequation.Ifnot,wedonothavealinearpattern.Inthiscase,wemaychoosetocontinuebycalculatingdifferencesofthefirstdifferences,whicharecalledseconddifferences,andsoonuntilwegetapatternwecanworkwith.

Intheexampleabove,wegetaconstantsetoffirstdifferences,whichtellsusthatthepatternisindeedlinear.

x‐value 0 1 2 3 4 5y‐value 6 9 12 15 18 21

FirstDifferences 3 3 3 3 3

ConvertingaLinearPatterntoanEquation

Creatinganequationfromthepatterniseasyifyouhaveconstantdifferencesanday‐valuefo scase,rx=0.Inthi

• Theequationtakestheform ,where• �m�istheconstantdifferencefromthetable,and

• �b�isthey‐valuewhenx=0.

Intheexampleabove,thisgivesustheequation: .

Finally,itisagoodideatotestyourequation.Forexample,if 4,theaboveequationgives 3 4 6 18,whichisthevalueinthetable.Sowecanbeprettysureourequationiscorrect.

Note:Ifthetabledoesnothaveavalueforx=0,youcanstillobtainthevalueof“b”.Simplyextendthetableleftorrightuntilyouhaveanx‐valueof0;thenusethefirstdifferencestocalculatewhatthecorrespondingy‐valuewouldbe.Thisbecomesyourvalueof“b”.

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ADVANCED

n ∆

n ∆ ∆2

n ∆ ∆2

n ∆ ∆2

IdentifyingNumberPatterns

Inthepatterntotheleft,noticethatthefirstandseconddifferencesappeartoberepeatingtheoriginalsequence.Whenthishappens,thesequencemayberecursive.Thismeansthateachnewtermisbasedonthetermsbeforeit.Inthiscase,theequationis:y n =y n‐1 +y n‐2 ,meaningthattogeteachnewterm,youaddthetwotermsbeforeit.

‐3‐11357

17 29

26 211

Whenlookingatpatternsinnumbers,isisoftenusefultotakedifferencesofthenumbersyouareprovided.Ifthefirstdifferencesarenotconstant,takedifferencesagain.

23

5 25

10 27

22222

37

52

7 24

11 48

19 816

35 1632

3 12

5 13

Inthepatterntotheleft,noticethatthefirstandseconddifferencesarethesame.Youmightalsonoticethatthesedifferencesaresuccessivepowersof2.Thisistypicalforan

exponentialpattern.Inthiscase,theequationis:y=2 x +3 .

Whenfirstdifferencesareconstant,thepatternrepresentsalinearequation.Inthiscase,theequationis:y=2x‐5 .Theconstantdifferenceisthecoefficientofxintheequation.

Whenseconddifferencesareconstant,thepatternrepresentsa

quadraticequation.Inthiscase,theequationis:y=x 2 +1 .The

constantdifference,dividedby2,givesthecoefficientofx2intheequation.

Algebra

8 25

13 38

21

Whentakingsuccessivedifferencesyieldspatternsthatdonotseemtolevelout,thepatternmaybeeitherexponentialorrecursive.

21

67

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n n

n ∆ ∆2 ∆3 n ∆ ∆2 ∆3

n ∆ ∆2 ∆3 n ∆ ∆2 ∆3

n ∆ ∆2 ∆3 n ∆ ∆2 ∆3

CompletingNumberPatternsAlgebra

Thefirststepincompletinganumberpatternistoidentifyit.Then,workfromtherighttotheleft,fillinginthehighestorderdifferencesfirstandworkingbackwards(left)tocompletethetable.Belowaretwoexamples.

‐17

6 1219 6

25 18

123214

625

‐17

6 1219

214

37

25 1837

662 24

61 6123 30

91

662 24

61 6123

Considerintheexamplesthesequencesofsixnumberswhichareprovidedtothestudent.Youareaskedtofindtheninthtermofeachsequence.

Example1 Example2

235

‐1

62 81321

21

3 12 0

5 13 1

8

12 0

5 13 1

25 1

13 38

21

35

8 25 1

13 38 2

21

Step1:Createatableofdifferences.Takesuccessivedifferencesuntilyougetacolumnofconstantdifferences(Example1)oracolumnthatappearstorepeatapreviouscolumnofdifferences(Example2).

Step2:Inthelastcolumnofdifferencesyoucreated,continuetheconstantdifferences(Example1)ortherepeateddifferences(Example2)downthetable.Createasmanyentriesasyouwillneedtosolvetheproblem.Forexample,ifyouaregiven6termsandaskedtofindthe9thterm,youwillneed3(=9‐6)additionalentriesinthelastcolumn.

‐1 27

24 8

6

21

3

6

3091 6

2146

6

3

261

16 12 3 1

19 6 2

Step3:Workbackwards(fromrighttoleft),fillingineachcolumnbyaddingthedifferencesinthecolumntotheright.

169 6 21

123 30 13 391

025 18 5 1

37 6 3 162

36 21 5127 6 13

8 2214

6 5 1

Columnn:214+127=341;341+169=510;510+217=727

Thefinalanswerstotheexamplesaretheninthitemsineachsequence,theitemsinboldred.

Intheexampletotheleft,thecalculationsareperformedinthefollowingorder:

Column∆2:30+6=36;36+6=42;42+6=48

Column∆:91+36=127;127+42=169;169+48=217 5510 48 55 13

217 34727 89

341 42 34 8

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AlgebraBasicNumberSets

NumberSet Definition Examples

NaturalNumbers(or,CountingNumbers)

Numbersthatyouwouldnormallycountwith.

1,2,3,4,5,6,�

WholeNumbers AddthenumberzerotothesetofNaturalNumbers

0,1,2,3,4,5,6,�

Integers WholenumbersplusthesetofnegativeNaturalNumbers

�‐3,‐2,‐1,0,1,2,3,�

RationalNumbers

Anynumberthatcanbeexpressed

intheform,whereaandbare

integersand .0 23

Allintegers,plusfractionsandmixednumbers,suchas:

, 176

, 345

RealNumbersAnynumberthatcanbewrittenindecimalform,evenifthatformisinfinite.

Allrationalnumbersplusrootsandsomeothers,suchas:

√2,√12 ,,e

BasicNumberSetTree

RealNumbers

Rational Irrational

Integers FractionsandMixedNumbers

Whole NegativeNumbers Integers

Natural ZeroNumbers

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AlgebraOperatingwithRealNumbers

AbsoluteValueTheabsolutevalueofsomethingisthedistanceitisfromzero.Theeasiestwaytogettheabsolut anum oelimin e ign.A l values ositiveor0.evalueof berist at itss bso ute arealwaysp

|5| 5 |3| 3 |0| 0

|1.5| 1.5

AddingandSubtractingRealNumbers

6 9 3 12 6 18

AddingNumberswiththeSameSign:

• Addthenumberswithoutregardtosign.

• Givetheanswerthesamesignastheoriginalnumbers.

• Examples:6 3 37 11 4

AddingNumberswithDifferentSigns:

• Ignorethesignsandsubtractthesmallernumberfromthelargerone.

• Givetheanswerthesignofthenumberwiththegreaterabsolutevalue.

• Examples:

6 3 3 6 3 13 4 13 4 9

SubtractingNumbers:

• Changethesignofthenumberornumbersbeingsubtracted.

• Addtheresultingnumbers.

• Examples:

MultiplyingandDividingRealNumbers

6 18 3 112 3 4 4

8

thesign

• Givetheanswera“+”sign.

NumberswiththeSameSign:

• Multiplyordivide numberswithoutregardto .

• Examples: 6 3 1812 3 4

NumberswithDifferentSigns:

• Multiplyordividethenumberswithoutregardtosign.

• Givetheanswera“‐”sign.• Examples:

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AlgebraPropertiesofAlgebra

PropertiesofAdditionandMultiplication.Foranyrealnumbersa,b,andc:

Property DefinitionforAddition DefinitionforMultiplication

ClosureProperty isarealnumber isarealnumber

IdentityProperty 0 0 1 1

InverseProperty 0 0, 1

1 1

CommutativeProperty

AssociativeProperty

DistributiveProperty

PropertiesofZero.Foranyrealnumbera:

Multiplicationby0 0 0 0

0DividedbySomething 0, 0

Divisionby0 is undeined even if a 0

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AlgebraPropertiesofAlgebra

OperationalPropertiesofEquality.Foranyrealnumbersa,b,andc:

Property Definition

AdditionProperty ,

SubtractionProperty ,

MultiplicationProperty ,

DivisionProperty 0,

OtherPropertiesofEquality.Foranyrealnumbersa,b,andc:

Property Definition

ReflexiveProperty

SymmetricProperty ,

TransitiveProperty ,

SubstitutionProperty If , then either can be substituted for the other in any equation (or inequality).

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AlgebraSolvingMulti‐StepEquations

ReversePEMDAS

Onesystematicwaytoapproachmulti‐stepequationsisReversePEMDAS.PEMDASdescribestheorderofoperationsusedtoevaluateanexpression.Solvinganequationistheoppositeofevaluatingit,soreversingthePEMDASorderofoperationsseemsappropriate.

Theguidingprinciplesintheprocessare:

• Eachstepworkstowardisolatingthevariableforwhichyouaretryingtosolve.

• Eachstep“un‐does”anoperationinReversePEMDASorder:

Subtraction Addition

Division Multiplication

Exponents Logarithms

Parentheses RemoveParentheses(andrepeatprocess)

Inverses

InversesNote:Logarithmsaretheinverseoperatortoexponents.ThistopicistypicallycoveredinthesecondyearofAlgebra.Inverses

Thelistaboveshowsinverseoperationrelationships.Inordertoundoanoperation,youperformitsinverseoperation.Forexample,toundoaddition,yousubtract;toundodivision,youmultiply.Hereareacoupleofexamples:

Example2

Solve: 2 2 5 53 Step1:Add3 3 3

Result: 2 2 5 2Step2:Divideby2 2 2

Result: 2 5 1Step3:Removeparentheses

Result: 2 5 1Step4:Subtract5 5 5

Result: 2 6Step5:Divideby2 2 2

Result: 3

Inverses

Example1

Solve: 3 4 14Step1:Add4 4 4

Result: 3 18Step2:Divideby3 3 3

Result: 6

Noticethatweaddandsubtractbeforewemultiplyanddivide.ReversePEMDAS.

Withthisapproach,youwillbeabletosolvealmostanymulti‐stepequation.Asyougetbetteratit,youwillbeabletousesomeshortcutstosolvetheproblemfaster.Sincespeedisimportantinmathematics,learningafewtipsandtrickswithregardtosolvingequationsislikelytobeworthyourtime.

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Example1

Solve: 8

Multiplyby:

Result: 8

12

Explanation:Sinceisthereciprocalof

,

whenwemultiplythem,weget1,and1 .Usingthisapproach,wecanavoiddividingbyafraction,whichismoredifficult.

Example2

Solve: 2

Multiplyby4: 4 4

Result: 2 4 8

Explanation:4isthereciprocalof ,so

whenwemultiplythem,weget1.Noticetheuseofparenthesesaroundthenegativenumbertomakeitclearwearemultiplyingandnotsubtracting.

Example3

Solve: 2 2 5 3 5Step1:Eliminateparentheses

Result: 4 10 3 5Step2:Combineconstants

Result: 4 7 5Step3:Subtract7 7 7

Result: 4 12Step4:Divideby4 4 4

Result: 3

AlgebraTipsandTricksinSolvingMulti‐StepEquations

FractionalCoefficients

Fractionspresentastumblingblocktomanystudentsinsolvingmulti‐stepequations.Whenstumblingblocksoccur,itisagoodtimetodevelopatricktohelpwiththeprocess.Thetrickshownbelowinvolvesusingthereciprocalofafractionalcoefficientasamultiplierinthesolutionprocess.(Rememberthatacoefficientisanumberthatismultipliedbyavariable.)

AnotherApproachtoParentheses

IntheReversePEMDASmethod,parenthesesarehandledafterallotheroperations.Sometimes,itiseasiertooperateontheparenthesesfirst.Inthisway,youmaybeabletore‐statetheprobleminaneasierformbeforesolvingit.

Example3,atright,isanotherlookattheprobleminExample2onthepreviouspage.

Usewhicheverapproachyoufindmosttoyourliking.Theyarebothcorrect.

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AlgebraProbabilityandOdds

Probability

Probabilityisameasureofthelikelihoodthataneventwilloccur.Itdependsonthenumberofoutcomesthatrepresentthee onterms,ventandthetotalnumberofpossibleoutcomes.Inequati

Example1:Theprobabilityofaflippedcoinlandingasaheadis1/2.Therearetwoequallylikelyeventswhenacoinisflipped�itwillshowaheadoritwillshowatail.So,thereisonechanceoutoftwothatthecoinwillshowaheadwhenitlands.

1

2 12

Example2:Inajar,thereare15bluemarbles,10redmarblesand7greenmarbles.Whatistheprobabilityofselectingaredmarblefromthejar?Inthisexample,thereare32totalmarbles,10ofwhicharered,sothereisa10/32(or,whenred bility sele tingaredmarble.uced,5/16)proba of c

10

32

1032

5

16

Odds

Oddsaresimilartoprobability,exceptthatwemeasurethenumberofchancesthataneventwilloccurrelativetothenumberofchancesthattheeventwillnotoccur.

Intheaboveexamples,

1 1

11

10

22

1022

5

11

• Notethatthenumeratorandthedenominatorinanoddscalculationaddtothetotalnumberofpossibleoutcomesinthedenominatorofthecorrespondingprobabilitycalculation.

• Tothebeginningstudent,theconceptofoddsisnotasintuitiveastheconceptofprobabilities;however,theyareusedextensivelyinsomeenvironments.

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AlgebraProbabilitywithDice

SingleDie

Probabilitywithasingledieisbasedonthenumberofchancesofaneventoutof6possibleoutcomesonthedie.Forexample:

2

5

TwoDice

Probabilitywithtwodiceisbasedonthenumberofchancesofaneventoutof36possibleoutcomesonthedice.Thefollowingtableofresultswhenrolling2diceishelpfulinthisregard:

1stDie

2ndDie 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Theprobabilityofrollinganumberwithtwodiceisthenumberoftimesthatnumberoccursinthetable,dividedby36.Herearetheprobabilitiesforallnumbers2to12.

2 5

8

11

3

6

9

12

4

7

10

3

4

6

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AlgebraCombinations

SingleCategoryCombinationsThenumberofcombinationsofitemsselectedfromaset,severalatatime,canbecalculatedrelativelyeasilyusingthefollowingtechnique:

Technique:Createaratiooftwoproducts.Inthenumerator,startwiththenumberoftotalitemsintheset,andcountdownsothetotalnumberofitemsbeingmultipliedisequaltothenumberofitemsbeingselected.Inthedenominator,startwiththenumberofitemsbeingselectedandcountdownto1.

Example:Howmanycombinationsof3itemscanbeselectedfromasetof8items?A nswer:

8 7 63 2 1

56

Example:Howmanycombinationsof4itemscanbeselectedfromasetof13item : s?Answer

13 12 11 104 3 2 1

715

Example:Howmanycombinationsof2itemscanbeselectedfromasetof30items? :Answer

30 292 1

435

MultipleCategoryCombinationsWhencalculatingthenumberofcombinationsthatcanbecreatedbyselectingitemsfromseveralcategories,thetechniqueissimpler:

Technique:Multiplythenumbersofitemsineachcategorytogetthetotalnumberofpossiblecombinations.

Example:Howmanydifferentpizzascouldbecreatedifyouhave3kindsofdough,4kindsofcheeseand8kindsoftoppings?Answer:

3 4 8 96

Example:Howmanydifferentoutfitscanbecreatedifyouhave5pairsofpants,8shirtsand4jackets?Answer:

5 8 4 160

Example:Howmanydesignsforacarcanbecreatedifyoucanchoosefrom12exteriorcolors,3interiorcolors,2interiorfabricsand5typesofwheels?Answer:

12 3 2 5 360

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AlgebraStatisticalMeasures

Statisticalmeasureshelpdescribeasetofdata.Adefinitionofanumberoftheseisprovidedinthetablebelow:

Concept Description Calculation Example1 Example2

DataSet Numbers 35,35,37,38,45 15,20,20,22,25,54

Mean AverageAddthevaluesand

dividethetotalbythenumberofvalues

35 5

35 37 38 45 38

15 16

8 22 22 25 54 26

Median(1) MiddleArrangethevaluesfromlowtohighandtakethe

middlevalue(1)37 21(1)

Mode MostThevaluethatappearsmostofteninthedata

set35 20

Range SizeThedifferencebetweenthehighestandlowestvaluesinthedataset

45�35=10 54�15=39

Outliers(2) OddballsValuesthatlookvery

differentfromtheothervaluesinthedataset

none 54

Notes:(1) Ifthereareanevennumberofvalues,themedianistheaverageofthetwomiddlevalues.InExample2,themedianis21,

whichistheaverageof20and22.(2) Thequestionofwhatconstitutesanoutlierisnotalwaysclear.Althoughstatisticiansseektominimizesubjectivityinthe

definitionofoutliers,differentanalystsmaychoosedifferentcriteriaforthesamedataset.

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AlgebraIntroduction to Functions

Definitions• A Relation isa relationship between variables, usually expressed asan equation.• In a typical x y equation, the Domain of a relation isthe set of x‐valuesfor which y‐

valuescan be calculated. For example, in the relation √ the domain is 0because these are the valuesof x for which a square root can be taken.

• In a typical x y equation, the Range of a relation is the set of y‐values that result for allvaluesof the domain. For example, in the relation √ the range is 0 becausethese are the valuesof y that result from all the valuesof x.

e doma• A Function isa relation in which each element in th in hasonly onecorresponding element in the range.

• A One‐to‐One Function isa function in which each element in the range isproduced byonly one element in the domain.

Function Tests in 2‐Dimensions

Vertical Line Test – If a vertical line passes through the graph of a relation in any two locations,it isnot a function. If it isnot possible to construct a vertical line that passes through the graphof a relation in two locations, it isa function.

Horizontal Line Test – If a horizontal line passes through the graph of a function in any twolocations, it isnot a one‐to‐one function. If it isnot possible to construct a horizontal line thatpasses through the graph of a function in two locations, it isa one‐to‐one function.

Examples:

Figure 1:

Not a function.

Failsvertical line test.

Figure 2:

Isa function, but not a one‐to‐one function.

Passesvertical line test.

Failshorizontal line test.

Figure 3:

Isa one‐to‐one function.

Passesvertical line test.Passeshorizontal line test.

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AlgebraSpecial Integer Functions

Greatest Integer FunctionAlso called the Floor Function, this function gives thegreatest integer lessthan or equal to a number. Thereare two common notationsfor this, asshown in theexamplesbelow.

Notation and examples:

3.5 3 2.7 3 6 6

2.4 2 7.1 8 0 0

In the graph to the right, notice the solid dotson the left of the segments (indicating the pointsareincluded) and the open lineson the right of the segments (indicating the pointsare not included).

Least Integer FunctionAlso called the Ceiling Function, this function gives theleast integer greater than or equal to a number. Thecommon notation for this isshown in the examplesbelow.

Notation and examples:

3.5 4 2.7 2 6 6

In the graph to the right, notice the open dotson theleft of the segments (indicating the pointsare not included) and the closed dotson the right of thesegments (indicating the pointsare included).

Nearest Integer FunctionAlso called the Rounding Function, thisfunction givesthe nearest integer to a number (rounding to the evennumber when a value ends in .5). There isno cleannotation for this, asshown in the examplesbelow.

Notation and examples:

3.5 4 2.7 3 6 6

In the graph to the right, notice the open dotson theleft of the segments (indicating the pointsare notincluded) and the closed dotson the right of the segments (indicating the pointsare included).

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AlgebraOperationswith Functions

Function Notation

Function notation replaces the variable y with a function name. The x in parenthesesindicatesthat x is the domain variable of the function. By convention, functionstend to use the letters f,g, and h asnamesof the function.

Operationswith Functions

2

1

1, 1

AddingFunctions

Subtracting Functions

Multiplying Functions

Dividing Functions , 0

The domain of the combinationof functions is the intersectionof the domainsof the twoindividual functions. That is,the combined function hasavalue in itsdomain if and only ifthe value is in the domain ofeach individual function.

Examples:

Let: Then:1

1

Note that in there isthe requirement 1. This isbecause 1 0 in the

denominator would require dividingby 0, producingan undefined result.

Other Operations

Other operationsof equa :lity also hold for functions, for example

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AlgebraComposition of Functions

In a Composition of Functions, first one function isperformed, and then the other. Thenotation for composition is, for example: or . In both of these notations,the function g isperformed first, and then the function f isperformed on the result of g.Alwaysperform the function closest to the variable first.

Double Mapping

A composition can be thought of asa double mapping. First g maps from itsdomain to itsrange. Then, f maps from the range of g to the range of f:

Example: Let

na d 1

Then:

And:

The WordsMethod

In the example,• The function sayssquare the argument.• The function saysadd 1 to the argument.

Sometimes it iseasier to think of the functions inather than in termsof an argument like x.words r

says“add 1 first, then square the result.”

says“square first, then add 1 to the result.”

Using the wordsmethod,

fg

Calculate: o 2

f: square it 2 4

g: add 1 to it 4 1

Calculate: o 12

g: add 1 to it 12 1

f: square it

Range of gDomain of fDomain of g Range of f

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AlgebraInversesof Functions

In order for a function to have an inverse, it must be a one‐to‐one function. The requirementfor a function to be an inverse is:

The notation isused for the ofInverse Function .

Another way of saying this is that if , then for all in the domain of .

Deriving an Inverse Function

The following stepscan be used to derive an inverse function. Thisprocessassumesthat theoriginal function isexpressed in termsof .

• Make sure the function isone‐to‐one. Otherwise it hasno inverse. You can accomplishthisby graphing the function applying the vertical and horizontal line tests.and

• Substitute the variable y for .• Exchange variables. That is, change all the x’s to y’sand all the y’s to x’s.• Solve for the new y in termsof the new x.• (Optional) Switch the e tion if you like.xpressionson each side of the equa• Replace the variable y with the function notation .• Check your work.

Examples:

o1

22 1 1

2

Derive the ver :in se of 2

Substitute for :

1

2 1

Exchange variables: 2 1

Add 1: 21

Divide by 2:

Switch sides:

Change Notation:

To check the result, note that:

o 3 6 313 2 6

Derive the ver :in se of

Substitute for :

2

2

Exchange variables: 2

Subtract 2: 2

Multiply by 3: 3 6

Switch sides: 3 6

Change Notation:

To check the result, note that:

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AlgebraTransformation – Translation

A Translation isa movement of the graph of a relation to a different location in the plane. Itpreserves the shape and orientation of the graph on the page. Alternatively, a translation canbe thought of as leaving the graph where it isand moving the axesaround on the plane.

In Algebra, the translationsof primary interest are the vertical and horizontal translationsof agraph.

Vertical Translation

Starting form:

Vertical Translation:

At each point, the graph of the translation is unitshigher orlower dependingon whether ispositive or negative. Theletter isused asa convention when movingup or down. Inalgebra, usually representsa y‐value of some importance.

Note:• A positive the graph up.shifts• A negative shifts the graph down.

Horizontal Translation

Starting form:

Horizontal Translation:

At each point, the graph of the translation is units tothe left or right depending on whether ispositive ornegative. The letter isused asa convention whenmoving left or right. In algebra, usually representsanx‐value of some importance.

Note:• A positive the graph to the left.shifts• A negative shifts the graph to the right.

For horizontal translation, the direction of movement of the graph iscounter‐intuitive; becareful with these.

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AlgebraTransformation – Vertical Stretch and Compression

A Vertical Stretch or Compression isa stretch or compression in the vertical direction, relativeto the x‐axis. It doesnot slide the graph around on the plane like a translation. An alternativeview of a vertical stretch or compression would be a change in the scale of the y‐axis.

Vertical Stretch

Starting form:

Vertical Stretch: , 1

At each point, the graph isstretched vertically by a factor of. The result isan elongated curve, one that exaggeratesall

of the featuresof the original.

Vertical Compression

Starting form:

Vertical Compression: , 1

At each point, the graph iscompressed vertically by afactor of . The result isa flattened‐out curve, one thatmutesall of the featuresof the original.

Note: The formsof the equationsfor vert ical stretch and vert icalcompression are the same. Theonly difference is the value of " ".

Value of " " in Result ing Curve

0 reflection

x‐axis

1 compression

original curve

1 stretch

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AlgebraTransformation – Horizontal Stretch and Compression

A Horizontal Stretch or Compression isa stretch or compression in the horizontal direction,relative to the y‐axis. It doesnot slide the graph around on the plane like a translation. Analternative view of a horizontal stretch or compression would be a change in the scale of the x‐axis.

Horizontal Stretch

Note: The formsof the equationsfor the horizontal stretch and thehorizontal compression are thesame. The only difference is thevalue of " ".

Starting form:

Horizontal Stretch: ,

At each point, the graph isstretched horizontallyby a factor of . The result isa widened curve, onethat exaggeratesall of the featuresof the original.

Horizontal Compression

Starting form:

Horizontal Compression: ,

At each point, the graph iscompressed horizontally by a

factor of . The result isa skinnier curve, one that mutesall of the featuresof the original.

Value of " " in Result ing Curve

0 reflection

horizontal line

1 stretch

original curve

1 compression

Note: For horizontal stretch and compression, the change in the graph caused by the valueof “b” is counter‐intuitive; be careful with these.

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AlgebraTransformation – Reflection

A Reflect ion isa “flip” of the graph acrossa mirror in the plane. It preservesthe shape thegraph but can make it look “backwards.”

In Algebra, the reflectionsof primary interest are the reflectionsacrossan axis in the plane.

X‐AxisReflection

Starting form:

x‐axisReflection:

Note the following:

• At each point, the graph isreflected across the x‐axis.

• The form of the transformation isthe same asa vertical stretch orcompression with .

• The flip of the graph over the x‐axis is, in effect, a verticaltransformation.

Y‐Axis Reflection

Starting form:

y‐axisReflection:

Note the following:

• At each point, the graph isreflected across the y‐axis.

• The form of the transformation isthe same asa horizontal stretchor compression with .

• The flip of the graph over the y‐axis is, in effect, a horizontaltransformation.

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AlgebraTransform

Starting form:

ations– Summary

For purposesof the following table, the variablesh and k are positive to make the formsmorelike what the student will encounter when solving problemsinvolving transformations.

Transformation Summary

Form of Transformation Result of Transformation

Vertical translation up k units.Vertical translation down k units.

Horizontal translation left h units.Horizontal translation right h its.un

, 1 Vertical stretch by a factor of ., 1 Vertical compression by a factor of .

, 1 Horizontal compression by a factor of .

, 1 Horizontal stretch by a factor of .

Reflection acrossthe x‐axis (vertical).Reflection acrossthe y‐axis (horizontal).

Transformationsbased on the valuesof “a” and “b” (stretches,compressions, reflections) can berepresented by these graphics.

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AlgebraBuildinga Graph with Transformations

The graph of an equation can be built with blocksmade up of transformations. Asan example,we will build the graph of 2 3 4.

Step 2: Translate 3 units tothe right to get equation:

Step 1: Start with the basicquadratic equation:

Step 3: Stretch vertically bya factor of 2 to get equation:

Step 4: Reflect over thex‐axis to get equation:

Step 5: Translate up 4units to get equation:

Final Result: Show the graphof the final equation:

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AlgebraSlope of a Line

The slope of a line tellshow fast it risesor fallsas it moves from left to right. If the slope isrising, the slope ispositive; if it is falling, the slope isnegative. The letter “m ”isoften used asthe symbol for slope.

The two most useful waysto calculate the slope of a line are discussed below.

Mathematical Definition of Slope

The definition isbased on two pointswithcoordinates , and , . The definition,then, is:

Comments:

• You can select any 2 pointson the line.

• A table such as the one at right can be helpful for doingyour calculations.

• Note that

implies that

.

So, it doesnot matter which point you assign asPoint 1and which you assign asPoint 2. Therefore, neither doesit matter which point is first in the table.

• It is important that once you assign a point asPoint 1 and another asPoint 2, that you usetheir coordinatesin the proper placesin the formula.

x‐v luea y‐value

Point 2

Point 1

Difference

Examples:

For the two lines in the figure above, we get the following:

Green Line: Red Line:

Green Line x‐value y‐value

Point A 1 4

Point C ‐3 ‐4

Difference 4 8

Red Line x‐value y‐value

Point D 4 ‐2

Point B ‐4 2

Difference 8 ‐4

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AlgebraSlope of a Line (cont’d)

Rise over Run

An equivalent method of calculating slope that ismorevisual is the “Rise over Run” method. Under thismethod, it helps to draw vertical and horizontal linesthat indicate the horizontal and vertical distancesbetween pointson the line.

The slop cae can then be lculated asfollows:

=

The rise of a line ishow much it increases (positive) or decreases (negative) between twopoints. The run ishow far the line movesto the right (positive) or the left (negative) betweenthe same two points.

Comments:

• You can select any 2 pointson the line.

• It is important to start at the same point in measuring both the rise and the run.

• A good convention is to alwaysstart with the point on the left and work your way to theright; that way, the run (i.e., the denominator in the formula) isalwayspositive. The onlyexception to this iswhen the run iszero, in which case the slope isundefined.

• If the two pointsare clearly marked as integerson a graph, the rise and run may actually becounted on the graph. Thismakesthe processmuch simpler than using the formula for thedefinition of slope. However, when counting, make sure you get the right sign for the slopeof the line, e.g., moving down as the line movesto the right isa negative slope.

Examples:

For the two lines in th o :e figure above, we get the foll wing

Green Line:

Notice how similar thecalculations in the examplesare under the two methodsof calculating slopes.Red Line:

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AlgebraSlopesof VariousLines

line isvertical

When you look at a line, youshould notice the followingabout itsslope:

• Whether it is0, positive,negative or undefined.

• If positive or negative,whether it is less than 1,about 1, or greater than 1.

The purpose of the graphsonthispage is to help you get a feelfor these things.

Thiscan help you check:

• Given a slope, whether youdrew the line correctly, or

• Given a line, whether youcalculated the slopecorrectly.

245

line issteep and going down3

12

line issteep and goingup

1line goesup at a 45⁰ angle

1line goesdown at a 45⁰ angle

3

17

line isshallow and going down

211

line isshallow and goingup

0line ishorizontal

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AlgebraVariousFormsof a Line

There are three formsof a linear equation which are most useful to the Algebra student, eachof which can be converted into the other two through algebraic manipulation. The ability tomove between forms isa very useful skill in Algebra, and should be practiced by the student.

Standard Form

Standa mplesrd Form Exa

3 2 6

2 7 14

The Stan r o a linear equation is:da d Form f

where A, B, and Care real numbersand Aand B are not both zero.Usually in this form, the convention is for A to be positive.

Why, you might ask, is this “Standard Form?” One reason is that this form iseasily extended toadditional variables, whereasother formsare not. For example, in four variables, the StandardForm would be: . Another reason isthat this form easily lends itselfto analysiswith matrices, which can be very useful in solving systemsof equations.

Slope‐Intercept FormSlope‐ plesIntercept Exam

3 6

34

14

The Slope‐Intercept Form of a linear equation is the one mostfamiliar ents. It is:to many stud

where m is the slope and b is the y‐intercept of the line (i.e., thevalue at which the line crossesthe y‐axis in a graph). m and b must also be real numbers.

Point‐Slope Form

The Point‐Slope Form of a linear equation is the one used least bythe student, but it can be very useful in certain circumstances. Inparticular, asyou might expect, it isuseful if the student isasked forthe equation of a line and isgiven the line’sslope and thecoordin e. The form of the equation is:

Point‐Slope Examples

3 2 4

7 523atesof a point on the lin

where m is the slope and , isany point on the line. One strength of this form is thatequations formed using different pointson the same line will be equivalent.

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AlgebraSlopesof Parallel and Perpendicular Lines

Parallel Lines

Two lines if their slopesare equal.are parallel

• In form, if the valuesof arethe same.

Example: 2 3 and2 1

• In Standard Form, if the coefficientsof andare proportiona ions.l between the equat

Example: 3 2 5 and46 7

• Also, if the linesare both vertical (i.e., theirslopesare undefin de ).

Example: and3 2

Perpendicular Lines

Two linesare perpendicular if the product of theirslopes is . That is, if the slopeshave differentsignsand tive inverses.are multiplica

• In form, the valuesofmultiply to get 1..

Example: and6 5

3

• In Standard Form, if you add the product ofthe x‐coefficients to the product of the y‐coefficientsand get zero.

Example: and4 6 43 2 5 because 4 3 6 2 0

• Also, if one line is isundefined) and one line ishorizontal (i.e., 0).vertical (i.e.,Example: and6

3

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AlgebraParallel, Perpendicular or Neither

The following flow chart can be used to determine whether a pair of linesare parallel,perpendicular, or neither.

yes

yes

no

no

Are theslopesof thetwo lines the

same?

First, put both linesin:form.

Is theproduct of

the twoslopes= ‐1?

Result: Thelinesareneither.

Result: Thelinesareparallel.

Result: The linesare

perpendicular.

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AlgebraParallel, Coincident or Intersecting

The following flow chart can be used to determine whether a pair of linesare parallel,coincident, or intersecting. Coincident linesare lines that are the same, even though they maybe expressed differently. Technically, coincident linesare not parallel because parallel linesnever intersect and coincident lines intersect at all pointson the line.

The intersection of the two lines is:

• For intersecting lines, the point of intersection.

• For parallel lines, the empty set, .

• For coincident lines, all pointson the line.

yesyes

nono

Are theslopesof thetwo lines the

same?

First, put both lines in: form.

Are the y‐intercepts ofthe two linesthe same?

Result: Thelinesare

coincident.

Result: Thelinesareparallel.

Result: Thelinesare

intersecting.

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AlgebraProperties of Inequality

For any real numbersa, b , and c:

Property Definit oni

AdditionProperty

,

,

SubtractionProperty

,

,

MultiplicationProperty

For ,0

,

,

For ,0

,

,

DivisionProperty

For 0,

,

,

For 0,

,

,

Note: all propertieswhich hold for “<” also hold for “≤”, and all propertieswhich hold for “>”also hold for “≥”.

There isnothing too surprising in these properties. The most important thing to be obtainedfrom them can be described as follows: When you multiply or divide an inequality by anegative number, you must “ flip” the sign. That is, “<” becomes“>”, “>” becomes“<”, etc.

In addition, it isuseful to note that you can flip around an entire inequality as long asyou keepthe “pointy” p the sign directed at the sam . Examples:art of e item

is the same as4 4

3 2 is the same as 3 2

One way to remember thisis that when you flip aroundan inequality, you must alsoflip around the sign.

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AlgebraGraphsof Inequalit iesin One Dimension

Inequalities in one dimension are generally graphed on the number line. Alternatively, if it isclear that the graph isone‐dimensional, the graphscan be shown in relation to a number linebut not specifically on it (examplesof thisare on the next page).

One‐Dimensional Graph Components

• The endpoint(s) – The endpointsfor the ray or segment in the graph are shown aseitheropen or closed circles.

o If the point is included in the solution to the inequality (i.e., if the sign is≤ or ≥), thecircle isclosed.

o If the point isnot included in the solution to the inequality (i.e., if the sign is< or >),the circle isopen.

• The arrow – If all numbers in one direction of the number line are solutionsto theinequality, an arrow points in that direction.

o For < or ≤ signs, the arrow pointsto the left ( ).

o For > or ≥ signs, the arrow pointsto the right ( ).

• The line – in a simple inequality, a line isdrawn from the endpoint to the arrow. If there aretwo endpoints, a line isdrawn from one to the other.

Examples:

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AlgebraCompound Inequalit ies in One Dimension

Compound inequalitiesare a set of inequalities that must all be true at the same time. Usually,there are two inequalities, but more than two can also form a compound set. The principlesdescribed below easily extend to caseswhere there are more than two inequalities.

Compound Inequalit ieswith the Word “AND”

An exam n qualitieswith the would be:ple of compound i e word “AND”

12 2 or 1 These are the same conditions,expressed in two different forms.(Simple Form) (Compound Form)

Graphically, “AND” inequalitiesexist at pointswhere the graphsof the individual inequalitiesoverlap. This is the “intersection” of the graphsof the individual inequalities. Below are twoexamplesof graphsof compound inequalitiesusing the word “AND.”

A typical “AND” example: The result is asegment that contains the pointsthat overlapthe graphsof the individual inequalities.

“AND” compound inequalitiessometimesresultin the empty set. Thishappenswhen nonumbersmeet both conditionsat the same time.

Compound Inequalit ieswith the Word “OR”

Graphically, “OR” inequalitiesexist at pointswhere any of the original graphshave points. Thisis the “union” of the graphsof the individual inequalities. Below are two examplesof graphsofcompound inequalitiesusing the word “OR.”

Atypical “OR” example: The result isa pair ofraysextending in opposite directions, with agap in between.

“OR” compound inequalitiessometimesresult inthe set of all numbers. Thishappenswhen everynumber meetsat least one of the conditions.

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AlgebraInequalities in Two Dimensions

Graphingan inequality in two dimensions involves the followingsteps:• Graph the underlyingequation.

• Make the line solid or dotted based on whether the inequality containsan “=” sign.o For inequalitieswith “<” or “>” the line isdotted.o For inequalitieswith “≤” or “≥” the line issolid.

• Determine whether the region containing the solution set isabove the line or below theline.

o For inequalitieswith “>” or “≥” the shaded region isabove the line.o For inequalitieswith “<” or “≤” the shaded region isbelow the line.

• Shade in the appropriate region.

Example:

Graph the solution set of the following system of inequality: 1

Step 1: Graph the underlyingequation.

Step 2: Determine whether the lineid or dotted:should be sol

1 the > sign doesnotcontain “=”, so the line isdotted

Step 3: Determine the region to beshaded based on the sign in theequation:

1 the > sign indicatesshading above the line

The solution set is the shaded area.

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Algebra

Graphsof Inequalit ies in Two Dimensions

Dashed LineBelow the Line

Dashed LineAbove the Line

Solid LineBelow the Line

Solid LineAbove the Line

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AlgebraAbsolute Value Functions

Equations

Graphsof equationsinvolving absolute valuesgenerally have a “V” pattern. Whenever you seea “V” in a graph, think “absolute value.” Ageneral equation for an absolute value function isofthe form:

| | | |where,• the sign indicates whether the graph opens up (“ ” sign) or down (“ “ sign).• | | is the absolute value of the slopesof the lines in the graph.• (h, k) is the location of the vertex (i.e., the sharp point) in the graph.

Examples:

Equation: | 1| 2 Vertex = 1,2

1; | slopes| 1 Graph opens up

Equation: | 2 1 | 3 Vertex = 1, 3

2; | slopes| 2 Graph opens up

Equation: 3

Vertex = , 3

; | slopes|

Graph opens down

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AlgebraAbsolute Value Functions(cont’d)

Inequalit ies

Since a positive number and a negative number can have the same absolute value, inequalitiesinvolvingabsolute valuesmust be broken into two separate equations. For example:

3 4The first new equation issimply the originalequation without the absolute value sign.

| 3| 4

Note: the English ispoor, but the mathiseasier to remember with this trick!

Equation 1

Solve: 43 Step 1: Add 3 3 3

Result: 7

Equation 2

Solve: 3 4Step 1: Add 3 3 3

Result: 1

3 4Sign that determinesuse of “AND” or “OR”

In the second new equation, two thingschange: (1) the sign flips, and (2) the value onthe right side of the inequality changes itssign.

At thispoint the absolute value problem hasconverted into a pair of compound inequalities.

Next, we need to know whether to use “AND” or “OR” with the results. To decide which wordto use, look at the sign in the inequality; then …

• Use the word “AND” with “ less thand” signs.• Use the word “OR” with “greator” signs.

The solution to the above absolute value problem, then, is the same as the solution to thefollowin t alities:g se of compound inequ

7 1 The solution set isall x in the range (‐1, 7)

Note: the solution set to thisexample isgiven in “ range” notation. When using thisnotation,• use parentheses ( ) whenever an endpoint isnot included in the solution set, and• use square brackets[ ] whenever an endpoint d in the solution set.is include• Alwaysuse parentheses ( ) with infinity signs( ∞ ∞ ).

The range: 6 2

Notation: 2,6

The range: 2

Notation: ∞ , 2

Examples:

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AlgebraSystemsof Equations

A system of equations isa set of 2 or more equations for which we wish to determine allsolutionswhich satisfy each equation. Generally, there will be the same number of equationsasvariablesand a single solution to each variable will be sought. However, sometimes there iseither no solution or there isan infinite number of solutions.

There are many methodsavailable to solve a system of equations. We will show three of thembelow.

Graphinga Solution

In the simplest cases, a set of 2 equations in 2 unknownscan be solved usinga graph. Asingleequation in two unknowns isa line, so two equationsgive us2 lines. The following situationsare possible with 2 lines:

• They will intersect. In thiscase, the point of intersection is the only solution.• They will be the same line. In thiscase, all pointson the line are solutions (note: this is

an infinite set).• They will be parallel but not the same line. In thiscase, there are no solutions.

Examples

Solution Set:

All pointson the line.Although the equations lookdifferent, they actuallydescribe the same line.

Solution Set:

The point of intersectioncan be read off the graph;the point (2,0).

Solution Set:

The empty set;these parallel lineswill never cross.

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AlgebraSystemsof Equations(cont’d)

Substitution Method

In the Substitution Method, we eliminate one of the variablesby substituting into one of theequations itsequivalent in termsof the other variable. Then we solve for each variable in turnand check the result. The steps sare illustrated in the example below.in thisproces

Example: Solve for x and y if:and: 2 .

Step 1: Review the two equations. Look for a variable that can be substituted from oneequation into the other. In thisexample, we see a single “y” in the first equation; this isa primecandidate for substitution.

We will substitute from the first equation for in the second equation.

Step 2: substitution.Perform the

becomes:

Step 3: l e uation for the single v e t.So ve the resulting q ariabl that is lef

Step 4: Substitute the known variable into one of the original equationsto solve for theremaini

After thisstep, the solution is tentatively identified as:, , meaning the point (3 , 1) .

ng variable.

Step 5: Check the result by substituting the solution into the equation not used in Step 4. If thesolution iscorrect, the result should be a true statement. If it isnot, you have made a mistakeand should o r work carefully.check y u

Since this isa true mathematicalstatement, the solution (3, 1) canbe accepted ascorrect.

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AlgebraSystemsof Equations(cont’d)

Elimination Method

In the Substitution Method, we manipulate one or both of the equationsso that we can addthem and eliminate one of the variables. Then we solve for each variable in turn and check theresult. This isan outstandingmethod for systemsof equationswith “ugly” coefficients. Thesteps in thisprocessare illustrated in the example below. Note the flow of the solution on thepage.

Example: Solve for x and y if:and: 2 .

Step 6: Check the result by substitutingthe solution into the equation not used inStep 5. If the solution iscorrect, theresult should be a true statement. If it isnot, you have made a mistake and shouldcheck your work.

2

Step 1: Re‐write the equations instandard form.

Step 2: Multiply each equation by a valueselected so that, when the equationsare added,a variable will be eliminated.

(Multiply by 2)

(Multiply by ‐1) 2

Step 5: Substitute the result intoone of the original equationsandsolve for the other variable.

U 2

Step 3: Add the resulting equations.

Step 4: Solve for the variable.

Since this isa true mathematical statement, thesolution (3, 1) can be accepted ascorrect.

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AlgebraSystemsof Equations(cont’d)

Classification of SystemsThere are two main classificationsof systemsof equations: Consistent vs. Inconsistent, andDependent vs. Independent.

Consistent vs. Inconsistent• Consistent Systemshave one or more solutions.

• Inconsistent Systemshave no solutions. When you try to solve an inconsistent set ofequations, you often get to a point where you have an impossible statement, such as“1 2.” This indicates that there isno solution to the system.

Dependent vs. Independent• Linearly Dependent Systemshave an infinite number of solutions. In Linear Algebra, a

system is linearly dependent if there isa set of real numbers(not all zero) that, whenthey are multiplied by the equations in the system and the resultsare added, the finalresult iszero.

• Linearly Independent Systemshave at most one solution. In Linear Algebra, a system islinearly independent if it isnot linearly dependent. Note: some textbooks indicate thatan independent system must have a solution. This isnot correct; they can have nosolutions (see the middle example below). For more on this, see the next page.

Examples

One SolutionConsistent

Independent

No SolutionInconsistentIndependent

Infinite SolutionsConsistentDependent

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ADVANCEDAlgebra

Linear Dependence

0

Linear dependence isa concept from Linear Algebra, and isvery useful in determining ifsolutions to complex systemsof equationsexist. Essentially, a system of functions isdefinedto e of real numbers (not all zero) such that:be linearly depend nt if there isa set ,

… 0 or, in summation notation,

If there isno set of real numbers , such that the above equationsare true, the system issaidto be linearly in e e n .d p nde t

The expression iscalled a linear combination of the functions . Theimportance of the concept of linear dependence lies in the recognition that a dependentsystem is redundant, i.e., the system can be defined with fewer equations. It isuseful to notethat a linearly dependent system of equationshasa determinant of coefficientsequal to 0.

Example:

Consider the fo w ions:llo ing system of equat

Notice that: .Therefore, the system is linearlydependent.

Check th th coeffi nt atrix:ing e determinant of e cie m

3 2 11 1 21 0 5

1 2 11 2

0 3 11 2

5 3 21 1

1 5 0 7 5 1 0.

It should be noted that the fact that D 0 is sufficient to prove linear dependence only if thereare no constant termsin the functions(e.g., if the problem involvesvectors). If there areconstant terms, it isalso necessary that these termscombine “properly.” There are additionaltechniques to test this, such asthe use of augmented matricesand Gauss‐Jordan Elimination.

Much of Linear Algebra concernsitself with setsof equationsthat are linearly independent. Ifthe determinant of the coefficient matrix isnon‐zero, then the set of equations is linearlyindependent.

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AlgebraSystemsof Inequalit ies in Two Dimensions

Systemsof inequalitiesare setsof more than one inequality. To graph a system of inequalities,graph each inequality separately (including shading in the appropriate region). The solution set,then, iseither the overlap of the regionsof the separate inequalities (“AND” Systems) or theunion of the regionsof the separate inequalities (“OR” Systems).

Examples:

Graph th t of the em of inequae solution se following syst lities:

(a) 2 3 AND 1 (b) 2 3 OR 1

Step 1: Graph the underlying equations.

Step 2: Determine whether each line should besolid or dotted: 2 3 the ≤ sign contains“=” , so the

line issolid

1 the > sign doesnot contain “=”,so the line isdotted

Step ne the regionsto be shaded based on the signs in the equations:3: Determi

• the ≤ sign indicatesshadingbelow the line2 3

• 1 the > sign indicatesshading above the line

Step 4: Determine the final solution set.

(a) If the problem hasan “AND” betweenthe inequalities, the solution set is theoverlap of the shaded areas(i.e., thegreen part in the graph below).

(b) If the problem hasan “OR” betweenthe inequalities, the solution set is theunion of all of the shaded areas(i.e.,the blue part in the graph below).

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AlgebraAlgebraParametric EquationsParametric Equations

Parametric Equations in 2 dimensionsare functions that expresseach of the two key variablesin termsof a one or more others. For exaParametric Equations in 2 dimensionsare functions that expresseach of the two key variablesin termsof a one or more others. For example,mple,

Parametric equationsare sometimesthe most useful way to solve a problem.Parametric equationsare sometimesthe most useful way to solve a problem.

Pythagorean TriplesPythagorean Triples

Asa xample, the followingparametric equationscan be used to find Pythagorean Triples:Asa xample, the followingparametric equationscan be used to find Pythagorean Triples:n e

Let , be relatively prime integersand let . Then, the following equationsproduce a setof integer values that satisf

n e

Let , be relatively prime integersand let . Then, the following equationsproduce a setof integer values that satisfy the Pythagorean Theorem:y the Pythagorean Theorem:

Examples:Examples:

ss tt aa bb cc Pytha r n shipPytha r n shipgo ean Relatiogo ean Relatio

3 2 5 12 13 5 12 13

4 3 7 24 25 7 24 25

5 2 21 20 29 21 20 29

5 3 16 30 34 16 30 34

Creatinga Standard Equation from Parametric Equations

To create a standard equation from a set ofparametricequations in two dimensions, Example: Create astandard equation for the

parametr eic quations:

r in the , we getSolving fo t :first equation

Substituting tion gives:into the second equa

ionCleaning thisup, we seek:seek:we get the solut

• Solve one parametric equation for t .• Substitute thisvalue of t into the other

equation.• Clean up the remaining expression as

necessary.

Note: any other method of solvingsimultaneousequationscan also be used forthispurpose.

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AlgebraExponent Formulas

W ord Descript ion of Property

Math Descript ion of Property

Limitat ions on variables

Examples

Product of Powers

Quotient of Powers

Power of a Power

Anything to the zero power is 1

, if , ,

Negative powers generate the reciprocal of what a posit ive power generates

Power of a product

Power of a quot ient

Convert ing a root to a power √ √

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AlgebraScientific Notation

Format

A number in scientific notation has two parts:

• A number which isat least 1 and is less than 10 (i.e., it must have only one digit beforethe decimal point). Thisnumber iscalled the coefficient.

• A power of 10 which ismultiplied by the first number.

Here are a f regular scientific n ifew examples o

3

numbersexpressed in

0.0 .

otat on.

132 .2 10

1,420,000 1.42 10

0034 3 4 10

1000 1 10

1 10

450 4.5 10

How many digits? How many zeroes?

There are a couple of simple rules for converting from scientific notation to a regular number orfor converting from a regular number to scientific notation:

• If a regular number is less than 1, the exponent of 10 in scientific notation isnegative.The number of leading zeroes in the regular number isequal to the absolute value ofthisexponent. In applying this rule, you must count the zero before the decimal point inthe regular number. Examples:

Original Number Action Conversion

0.00034 Count 4 zeroes 3.4 x 10‐4

6.234 x 10‐8 Add 8 zeroesbefore the digits 0.000 000 062 34

• If the number isgreater than 1, the number of digitsafter the first one in the regularnumber isequal to the exponent of 10 in the scientific notation.

Original Number Action Conversion

4,800,000 Count 6 digitsafter the “4” 4.8 x 106

9.6 x 103 Add 3 digitsafter the “9” 9,600

• Asa general rule, multiplying by powersof 10 moves the decimal point one place foreach power of 10.

o Multiplying by positive powersof 10 movesthe decimal to the right.o Multiplying by negative powersof 10 moves the decimal to the left.

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AlgebraAddingand Subtractingwith Scientific Notation

When adding or subtracting numbersin scientific notation:

• Adjust the numbersso they have the same power of 10. Thisworksbest if you adjustthe representation of the smaller number so that it has the same power of 10 as thelarger number. To do this:

o Call the difference between the exponentsof 10 in the two numbers“n”.o Raise the power of 10 of the smaller number by “n”, ando Move the decimal point of the coefficient of the smaller number “n” places to

the left.• Add the coefficients, keeping the power of 10 unchanged.

• If the result isnot in scientific notation, adjust it so that it is.o If the coefficient isat least 1 and lessthan 10, the answer is in the correct form.o If the coefficient is10 or greater, increase the exponent of 10 by 1 and move the

decimal point of the coefficient one space to the left.o If the coefficient is less than 1, decrease the exponent of 10 by 1 and move the

decimal point of the coefficient one space to the right.

Examples:

3.2 10 0.32 10

9.9 10 9 9. 0 10

10. 22 10

1.022 10

Explanation: Aconversion of the smallernumber isrequired prior to adding because theexponentsof the two numbersare different.After adding, the result isno longer in scientificnotation, so an extra step isneeded to convert itinto the appropriate format.

6.1 10 6.1 10

2.3 10 2.3 10

8. 4 10

1.2 10 1.20 10

4.5 10 0.45 10

0.75 10

7.5 10

Explanation: No conversion isnecessarybecause the exponentsof the two numbersarethe same. After adding, the result is in scientificnotation, so no additional stepsare required.

Explanation: Aconversion of the smallernumber isrequired prior to subtracting becausethe exponentsof the two numbersare different.After subtracting, the result isno longer inscientific notation, so an extra step isneeded toconvert it into the appropriate format.

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AlgebraMultiplyingand Dividingwith Scientific Notation

When multiplying or dividing numbers in scientific notation:

• Multiply or divide the coefficients.

• Multiply or divide the powersof 10. Remember that thismeansaddingor subtractingthe exponentswhile keeping the base of 10 unchanged.

o If you are multiplying, add the exponentsof 10.o If you are dividing, subtract the exponentsof 10.

• If the result isnot in scientific notation, adjust it so that it is.o If the coefficient isat least 1 and lessthan 10, the answer is in the correct form.o If the coefficient is10 or greater, increase the exponent of 10 by 1 and move the

decimal point of the coefficient one space to the left.o If the coefficient is less than 1, decrease the exponent of 10 by 1 and move the

decimal point of the coefficient one space to the right.

Examples:

4 10

5 10

Explanation: The coefficientsare multiplied andthe exponentsare added. After multiplying, theresult isno longer in scientific notation, so anextra step isneeded to convert it into theappropriate format.

20 10

2.0 10

1.2 10

2.0 10

2. 4 10

Explanation: The coefficientsare multiplied andthe exponentsare added. After multiplying, theresult is in scientificnotation, so no additionalstepsare required.

3.3 10 Explanation: The coefficientsare divided andthe exponentsare subtracted. After dividing,the result isno longer in scientific notation, soan extra step isneeded to convert it into theappropriate format.

5.5 10

0.6 10

6.0 10

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AlgebraIntroduction to Polynomials

What isa Polynomial?

A polynomial isan expression that can be written asa term or a sum of terms, each of which isthe product of a scalar (the coefficient) and a seriesof variables. Each of the terms isalso calleda monomial.

Examples (all of these are polynomials):

Monomial 3 4

Binomial 2 8 15 12

Trinomial 6 9 7 3

Other 4 6 4 1 2 6 3 8 2

Definitions:

Scalar: A real number.

Monomial: Polynomial with one term.

Binomial: Polynomial with two terms.

Trinomial: Polynomial with three terms.

Degree of a Polynomial

The degree of a monomial is the sum of the exponentson itsvariables.

The degree of a polynomial is the highest degree of any of itsmonomial terms.

Examples:

Polynomial Degree P aolynomi l Degree

6 0 3 6 6

3 1 15 12 9

3 3 7 3 5

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AlgebraAddingand SubtractingPolynomials

Problemsasking the studen written in linear form:t to add or subtract polynomialsare often

Add: 3 2 4 2 4 6

The problem ismuch more easily solved if the problem iswritten in column form, with eachpolynomial written in standard form.

Definitions

Standard Form: A polynomial in standard form has its termswritten from highest degree tolowest degree from left to right.

Example: The standard form of 3 4 is 3 4

Like Terms: Termswith the same variablesraised to the same powers. Only the numericalcoefficientsare different.

Example: 2 , 6 , and are like terms.

Addition and Subtraction Steps

Step 1: Write each polynomial in standard form. Leave blank spaces for missing terms. Forexample, if adding 3 2 4 , leave space for the missing ‐term.

Step 2: If you are subtracting, change the sign of each term of the polynomial to be subtractedand add instead. Adding ismuch easier than subtracting.

Step 3: Place the polynomials in column form, being careful to line up like terms.

Step 4: Add the polynomials.

Examples:

: 3 2 4 2 4 6

3 2 4

2 4 6

3 2 6 2

:

Solution :

3 2 4 2 4 6

3 2 4

2 4 6

3 2 2 10

Solution:

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AlgebraMultiplyingBinomials

The three methodsshown below are equivalent. Use whichever one you like best.

FOILMethod

FOILstands for First, Outside, Inside, Last. To multiply using the FOILmethod, you make fourseparate multiplicat a lts.ionsand dd the resu

Example: Mult y 3 4ipl 2 3

First:

2 3 3 9 124 6 8

6 12

The result isobtained by adding the resultsofthe 4 separate multiplications.

F O I L2 3 6

Outside: 82 4

Inside: 93 3

Last: 3 4 12

Box Method

The Box Method ispretty much the same as the FOILmethod, except for the presentation. Inthe box method, a 2x2 array of multiplications iscreated, the 4 multiplicationsare performed,and the resultsare added.

Example: Multiply 2 3 3 4

Multip ly 3x

2x 6 8

+3 9 12

2 3 3 9 124 6 8

6 12

The result isobtained by adding the resultsofthe 4 separate multiplications.

Stacked Polynomial Method

2 33 4

1286 96 12

A third method is to multiply the binomialslike you would multiply 2‐digit numbers.The name comesfrom how the twopolynomialsare placed in a “stack” inpreparation for multiplication.

Example: Multiply 2 3 3 4

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AlgebraMultiplyingPolynomials

If the polynomialsto be multiplied contain more than two terms(i.e., they are larger thanbinomials), the FOILMethod will not work. Instead, either the Box Method or the StackedPolynomial Method should be used. Notice that each of these methods isessentially a way toapply the distributive property of multiplication over addition.

The methodsshown below are equivalent. Use whichever one you like best.

Box Method

The Box Method is the same for larger polynomialsas it is for binomials, except the box isbigger. An array of multiplications iscreated; the multiplicationsare performed; and like termsare added.

Example: Multiply 2 2 3 2 3 4

Mu plylti

2 3 4

4 6 8

4 6 8

6 9 12

42 2 3 2 3

2

4 3

64 4

6 6 8

9 8

Results:

Stacked Polynomial MethodResults:

2 2 3

32 4

124 8 8

6 93 6

42 4 6

2 7 6 8 17 12

In the Stacked Polynomial Method, thepolynomialsare multiplied using the sametechnique to multiply multi‐digit numbersOne helpful tip is to place the smallerpolynomial below the larger one in thestack.

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AlgebraAlgebraDividingPolynomialsDividingPolynomials

Dividing polynomials isperformed much like dividing large numbers long‐hand.Dividing polynomials isperformed much like dividing large numbers long‐hand.

Long Division MethodLong Division Method

ThisproceThisprocess isbest described by example:

Example: 2 5 2 2

ss isbest described by example:

Example: 2 5 2 2

2 2 5 2

2

2 2 5 2

2

2 22 52 4

2

2 1

2 22 52 4

2

Step 1: Set up the division like a typical long handdivision problem.Step 1: Set up the division like a typical long handdivision problem.

Step 2: Divide the leading term of the dividend bythe leading term of the divisor. Place the result

f like degree of the dividend.

Step 2: Divide the leading term of the dividend bythe leading term of the divisor. Place the result

f like degree of the dividend.above the term o

2 2

above the term o

2 2

Step 3: Multiply the new term on top by the divisorand subtract fromStep 3: Multiply the new term on top by the divisorand subtract from the dividend.

2 2 2 4

the dividend.

2 2 2 4

Step 4: Repeat steps2 and 3 on the remainder ofthe division until the problem iscompleted.Step 4: Repeat steps2 and 3 on the remainder ofthe division until the problem iscompleted.

Thisprocess results in the final answer appearingaThisprocess results in the final answer appearinga

2bove the dividend, so that:

2 5 2 2 2 1

bove the dividend, so that:

2 5 2 2 2 1220r0

RemaindersRemaindersIf there were a remainder, it would be appended tothe result of the problem in the form of a fraction, just like when dividing integers. Foexample, in the problem above, if the remainder were 3, the fraction

If there were a remainder, it would be appended tothe result of the problem in the form of a fraction, just like when dividing integers. Forexample, in the problem above, if the remainder were 3, the fraction would be added to

the result of the division. 2 5 1 2 2 1

AlternativesThisprocesscan be tedious. Fortunately, there are better methodsfor dividingpolynomialsthan longdivision. These include Factoring, which isdiscussed next and elsewhere in thisGuide, and Synthetic Division, which isdiscussed in the chapter on Polynomials– Intermediate.

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AlgebraFactoringPolynomials

Polynomialscannot be divided in the same way numberscan. In order to divide polynomials, itisoften useful to factor them first. Factoring involvesextracting simpler termsfrom the morecomplex polynomial.

Greatest Common Factor

The Greatest Common Factor of the termsof a polynomial isdetermined as follows:

Step 1: Find the Greatest Common Factor of the coefficients.Step 2: Find the Greatest Common Factor for each variable. This issimply each variable taken

to the lowest power that existsfor that variable in any of the terms.

Step 3: Multiply the GCFof the coefficientsby the GCFfor each variable.

GCF 18,42,30 6GCF , ,GCF , ,1 1GCF , ,

So, GCF polynomial 6

Example:

Find the GCFof 18 42 30

The GCFof the coefficientsand each variable are shownin the box to the right. The GCFof the polynomial is theproduct of the four individual GCFs.

Factoring Steps

Step 1: Factor out of all termsthe GCFof the polynomial.Note: Typically onlysteps1 and 2 areneeded in high schoolalgebraproblems.

Step 2: Factor out of the remaining polynomial any binomials that can beextracted.

Step 3: Factor out of the remaining polynomial any trinomials that canbe extracted.

Step 4: Continue thisprocessuntil no further simplification ispossible.

Examples:

Factor: 3 18 27

93 6

3 3

The factoringof the blue trinomial (2nd line) intothe square of a binomial is the result ofrecognizing the special form it represents. Specialformsare shown on the next two pages.

Factor: 6 24

6 4

6 2 2

The factoringof the blue binomial (2nd line) intobinomialsof lower degree is the result ofrecognizing the special form it represents. Specialformsare shown on the next two pages.

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AlgebraSpecial Formsof Quadratic Functions

It ishelpful to be able to recognize acouple special formsof quadratic functions. In particular,if you can recognize perfect squaresand differencesof squares, your work will become easierand more accurate.

Perfect SquaresPerfect squaresare of the form:

Identificat ion and Solution

The following stepsallow the student to identify and solve a trinomial that isa perfect square:Step 1: Notice the first term of the trinomial isa square. Take itssquare root.Step 2: Notice the last term of the trinomial isa square. Take itssquare root.Step 3: Multiply the resultsof the first 2 stepsand double that product. If the result is the

middle term of the trinomial, the expression isa perfect square.Step 4: The binomial in the solution is the sum or difference of the square rootscalculated in

steps1 and 2. The sign between the termsof the binomial is the sign of the middleterm of the trinomial.

–Example:

√

Notice that the middle term isdouble the productof the two square roots( and ). This isatelltale sign that the expression isa perfect square.

Identify the trinomial asa perfect square:• the of the first and last rms. They are 2 and 3 .Take square roots te• Test the middle term. Multiply the roots from the previousstep, then double the result:

2 3 2 12 . The result (with a “ ” sign in front) is the middle term of theoriginal trinomial. Therefore, the expression isa perfect square.

To expressthe trinomial as the square of a binomial:• The square rootsof the first and last terms 2 and 3 make the binomial we seek.up• We may choose the gn of the first term, so let’sc oose the “ ” sign.si h• Having chosen the “ ” sign for the first term, the second term of the binomial takesthe

sign of the middle term trinomial refore, the result is:of the original (“ ”). The

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AlgebraSpecial Formsof Quadratic Functions

Differencesof Squares

Differencesof squaresare of the form:

These are much easier to recognize than the perfect squaresbecause there isno middle termto cons isider. Notice why there no middle term:

these twotermscancel

Identificat ion

To see if an expression isa difference of squares, you must answer “yes” to four questions:1. Are there only two terms?2. Is there a “ ” sign between the two terms?3. Is the first term a square? If so, take itssquare root.4. Is the second term a square? If so, take itssquare root.

The solution isthe product of a) the sum of the square roots in questions3 and 4, and b) thedifference of the square roots in steps3 and 4.

Note: A telltale sign of when an expression might be the difference of 2 squares iswhen thecoefficientson the variablesare squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, etc.

Examples:(1)

(2)

(3)

(4)

ADVANCED: Over the field of complex numbers, it isalso possible to factor the sum of 2 squares:

This isnot possible over the field of real numbers.

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coefficientof x

sign 1 sign 2

constant

Example: Factor

The numberswe seek are4 and 7 because:

4 · 7 28, and

·

·

·

4 7 3

·

7 · 4

4 7 28

3 28

AlgebraFactoringTrinomials– Simple Case Method

A common problem in Elementary Algebra is the factoring of a trinomial that isneither aperfect square nor a difference of squares.

Consider the simple case where the coefficient of is1. The general form for thiscase is:

·

In order to simplify the illustration of factoring a polynomial where the coefficient of is1, wewill use the orange descriptorsabove for the componentsof the trinomial being factored.

Simple Case Method

Step 1: Set up parenthesesfor a pair of binomials. Put “x” in theleft hand position of each binomial.

Step 2: Put sign 1 in the middle position in the left binomial.

Step 3: Multiply sign 1 and sign 2 to get the sign for the rightbinomial. Remember:

· ·· ·

Step 4: Find two numbers that:

(a) Multiply to get the constant and,(b) Add to get the coefficient of

Step 5: Place the numbers in the binomialsso that their signsmatch the signsfrom Steps2 and 3. This is the finalanswer.

Step 6: Check your work by multiplying the two binomialsto seeif you get the original trinomial.

Fill in:___ · ___ ______ ___ ___

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Example: Factor

6 2

12

4 · 3 12

4 3 1

6 4 3 2

6 4 3 2

2 3 2 1 3 2

·

2 1 · 3 26 4 3 2

6 2

AlgebraFactoringTrinomials– ACMethod

There are timeswhen the simple method of factoring a trinomial isnot sufficient. Primarily thisoccurswhen the coefficient of isnot 1. In thiscase, you may use the ACmethod presentedhere, or you may use either the brute force method or the quadratic formula method(described on the next couple of pages).

ACMethod

The ACMethod derives itsname from the first step of theprocess, which is to multiply the valuesof “ ” and “ ” from thegeneral form of the quadratic equation:

Step 1: Multiply the valuesof “ ” and “ ”.

Step 2: Find two numbers that:

(a) Multiply to get the value of ,and

(b) Add to get the coefficient of

Step 3: Split the middle term into two terms, with coefficientsequal to the valuesfound in Step 2. (Tip: if only one ofthe coefficients isnegative, put that term first.)

Step 4: Group the terms into pairs.

Step 5: Factor each pair of terms.

Step 6: Use the distributive property to combine themultipliersof the common term. This is the finalanswer.

Step 7: Check your work by multiplying the two binomialstosee if you get the original trinomial.

Fill in:___ · ___ ___

___ ___ ___

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AlgebraFactoringTrinomials– Brute Force Method

When the coefficient of isnot 1, the factoring processbecomesmore difficult. There are anumber of methods that can be used in thiscase.

If the question being asked is to find rootsof the equation, and not to factor it, the student maywant to use the quadratic formula whenever the coefficient of isnot 1. Even if you arerequired to factor, and not just find roots, the quadratic formula may be a viable approach.

Brute Force Method

Example: Factor

which equat

Combinations that produce a productof 4 are:

1 and 4 or 2 and 2

Combinations that produce a productof 3 are:

1 and 3 or 1 a d 3n

1 4 31 4 33 13 1

44

2 1 2 312 2 3

1 4 3 4 31 4 3 4 3

3 4 1 4 11 33 4 1 4 11 3

2 1 2 3 4 4 342 1 2 3 4 3

4 4 3

Thismethod isexactly what it soundslike. Multiple equationsare possible and you must tryeach of them until you find the one that works. Here are the steps to finding ionsare candidate solutions:

Step 1: Find all setsof whole numbersthat multiply toget the coefficient of the first term in thetrinomial. If the first term ispositive, you needonly consider positive factors.

Step 2: Find all setsof whole numbersthat multiply toget the coefficient of the last term in thetrinomial. You must consider both positive andnegative factors.

Step 3: Create all possible productsof binomials thatcontain the whole numbersfound in the firsttwo steps.

Step 4: Multiply the binomial pairsuntil you find onethat results in the trinomial you are trying tofactor.

Step 5: Identify the correct solution.

Notice the patterns in the candidate solut ions in Step 4. Each pair of equations is identical except forthe sign of the middle term in the product. Therefore, you can cut your work in half by consideringonlyone of each pair until you see a middle term coefficient that has the right absolute value. If you haveeverything right but the sign of the middle term, switch the signs in the binomials to obtain the correctsolution. Remember to check your work!

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AlgebraFactoringTrinomials– Quadratic Formula Method

Quadratic Formula Method

The Quadratic Formula isdesigned specifically to find rootsof a second degree equation.However, it can also be used asa back‐door method to factor equationsof second degree. Thestepsare:

Step 1: Apply the quadratic formula the rootsof the equation.to determine

Step 2: Put each root into m: 0.the for

Step 3: Show the two binomialsasa product. Note that these binomialsmaycontain fractions. We will eliminate the fractions, if p sible, in the next step.os

Step 4: Multiply the binomials in tep 3 by the coefficient of the following way:S

(a) Break the coefficient of into itsprime factors.

(b) Allocate the prime factorsto the binomials in a way that eliminates the fractions.

Step 5: Check your work.

Example:

Factor:

Step 1: √ √ or

Step 2: t nscontaining rootsare:The two equa io 0 and 0.

Step 3:

Step 4: The coefficient of in the original equation is4, and 4 2 2. An inspection of thebinomials in Step 3 indicateswe need to multiply each binomial by 2 in order toeliminate the fractions:

2 2 3 and 2 2 1

So that: 4 4 3 in factored form

Step 5: Check (usingFOIL) 2 3 2 1 4 2 6 3 4 4 3,which is the equation we were trying to factor.

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AlgebraAlgebraSolvingEquationsby FactoringSolvingEquationsby Factoring

There are a number of reasons to factor a polynomial in algebra; one of the most commonreasonsis to find the zeroesof the polynomial. A “zero” isa domain value (e.g., x‐value) forwhich the polynomial generatesa value of zero. Each zero isa solution of the polynomial.

There are a number of reasons to factor a polynomial in algebra; one of the most commonreasonsis to find the zeroesof the polynomial. A “zero” isa domain value (e.g., x‐value) forwhich the polynomial generatesa value of zero. Each zero isa solution of the polynomial.

In facto polynomial’szeroes. Consider the following:In facto polynomial’szeroes. Consider the following:red form, it ismuch easier to find a

2 4 8 3 is the factored form of a polynomial.

red form, it ismuch easier to find a

2 4 8 3 is the factored form of a polynomial.

If a number of itemsare multiplied together, the result iszero whenever any of the individualitemsiszero. This is true for constantsand for polynomials. Therefore, if any of the factorsofthe polynomial hasa value of zero, then the whole polynomial must be zero. We use thisfactto find zeroesof polynomials in factored form.

If a number of itemsare multiplied together, the result iszero whenever any of the individualitemsiszero. This is true for constantsand for polynomials. Therefore, if any of the factorsofthe polynomial hasa value of zero, then the whole polynomial must be zero. We use thisfactto find zeroesof polynomials in factored form.

Example 1:Example 1:

Find the zeroesof 2 4 8 3 .Find the zeroesof 2 4 8 3 .

Step 1: Set theStep 1: Set the equation equal to zero.

2 4 8 3 0

equation equal to zero.

2 4 8 3 0

Step 2: The wh iszero whenever any of its factors iszero. For the example, thisoccurs

Step 2: The wh iszero whenever any of its factors iszero. For the example, thisoccurs

ole equationwhen:

or

ole equationwhen:

or2 0,

or

2 0,

or4 0,

or

4 0,

or

The solution set, then, is:

ventionally, the x‐valu

2, 4,8, , 3

or, more con esare putin numerical largest:

8 0,

or

8 0,

or0,

3 0

0,

3 0

order from smallest to

4, 3,2, , 8

The solution set contains the twodomain values that make the originalequation zero, namely:

1,61,6

Example 2:Example 2:

Find the zeroesofFind the zeroesof 7 6

0

7 6

07 6

6 1 0

7 6

6 1 0

0 00 06 1

6 1

6 1

6 1

Set Notation: We may list the setof solutions to aproblem byplacing the solutions in braces{},separated by commas.

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AlgebraIntroduction to QuadraticFunctions

Standard Form of a Quadratic Function

The Standard a l is:Form of second degree polynomia

with 0

An equation of this form iscalled a quadratic function.

The graph of thisequation iscalled a parabola.

Up or down?The direction in which the parabola openson a graph isbased on n ( or ) of in the equation.the sig

• If the parabola pointsdown and it opensup.0,

• If 0, the parabola pointsup and it opensdown.

If you forget thisrule, just remember that “up or down”dependson the sign of , and do a quick graph of ,where 1 on your paper.

Vertex and Axisof Symmetry

In Standard Form, the vertex of the parabola hascoordinates: , where “y” iscalculated

by substituting for “x” in the equation. The vertex iseither the highest point on the graph

(called a maximum) or the lowest point on the graph (called a minimum). It also lieson the axisof symmetry o h.f the grap

The equation 2 is called the axisof symmetry of the parabola.

Vertex Form of a Quadratic Function

A second usef ertex Form:ul form of a quadratic function isbased on the vertex, and iscalled V

where h,k is the vertex of the parabola

It ispossible to convert from Standard Form to Vertex Form and from Vertex Form to StandardForm. Both are equally correct.

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AlgebraCompleting the Square

A very useful method for solving quadratic equationsisCompleting the Square. In fact, this isthe method used to derive the quadratic formula from the general quadratic equation inStandard Form. The stepsinvolved in Completing the Square and an example are providedbelow:

Consider the startingequation:

Step 1: Modify the equation so that the coefficient of is1. To do this, simply divide thewhole equation by the value of “ ”.

Example: Consider t e equation: 3h 18 21 0

Divide by 3 to get: 6 7 0

Step 2: Get rid of the pesk constant. We will gene ey rat our own.

Example: Add 7 to both sides: 6 7

Step 3: Calculate a new constant. The required constant is the square of one half of thecoefficient of Add th sidesof the equation.. it to bo

Example: 6 73, 3 9.uare the result:Half it, then sq

Result: 6 9 7 9

Step 4: Recognize the left hand side of the equation asa perfect square. After all, that was thereason we sele onstant the way we did.cted the new c

Example: 3 16

Step 5: Take the squa . Remember the “ ” sign on the constant term.re root of both sides

Example: 3 √163 4

Step 6: Break the resu ion into two separ s, and solve.ltingequat ate equation

Example: 3 4 3 471

Solution: ,

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AlgebraTable of Powersand Roots

Squar e Root Number Squar e Cube 4 th Power

√1 1.000 1 1 1 1 1 1 1

√2 1.414 2 2 4 2 8 2 16

√3 1.732 3 3 9 3 27 3 81

√4 2.000 4 4 16 4 64 4 256

√5 2.236 5 5 25 5 125 5 625

√6 2.449 6 6 36 6 216 6 1,296

√7 2.646 7 7 49 7 343 7 2,401

√8 2.828 8 8 64 8 512 8 4,096

√9 3.000 9 9 81 9 729 9 6,561

√10 3.162 10 10 100 10 1,000 10 10,000

√11 3.317 11 11 121 11 1,331 11 14,641

√12 3.464 12 12 144 12 1,728 12 20,736

√13 3.606 13 13 169 13 2,197 13 28,561

√14 3.742 14 14 196 14 2,744 14 38,416

√15 3.873 15 15 225 15 3,375 15 50,625

√16 4.000 16 16 256 16 4,096 16 65,536

√17 4.123 17 17 289 17 4.913 17 83,521

√18 4.243 18 18 324 18 5,832 18 104,976

√19 4.359 19 19 361 19 6,859 19 130,321

√20 4.472 20 20 400 20 8,000 20 160,000

√21 4.583 21 21 441 21 9,261 21 194,481

√22 4.690 22 22 484 22 10,648 22 234,256

√23 4.796 23 23 529 23 12,167 23 279841

√24 4.899 24 24 576 24 13,824 24 331,776

√25 5.000 25 25 625 25 15,625 25 390,625

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AlgebraThe Quadratic Formula

The Quadratic Formula isone of the first difficult math formulas that studentsare asked tomemorize. Mastering the formula, though difficult, is full of rewards. By knowingwhy it worksand what the variouspartsof the formula are, a student can generate a lot of knowledge in ashort period of time.

For a quadratic function of the form:

The formula for the roots (i.e., where y = 0) is: Quadratic

Formula

How Many Real Roots?

The discriminant is the part under the radical:

• If the discriminant isnegative, the quadratic function has0 real roots. This isbecause anegative number under the radical results in imaginary roots instead of real roots. Inthiscase the graph the graph will not cross the x‐axis. It will be either entirely above thex‐axisor entirely below the x‐axis, depending on the value of “a”.

• If the discriminant iszero, the quadratic function has1 real root. The square root of

zero iszero, so the radical disappearsand the only root is 2 . In thiscase, the

graph will appear to bounce off the x‐axis; it touches the x‐axisat only one spot – thevalue of the root.

• If the discriminant ispositive, the quadratic function has2 real roots. This isbecause areal square root exists, and it must be added in the formula to get one root andsubtracted to get the other root. In thiscase, the graph will cross the x‐axis in twoplaces, the valuesof the roots.

Where are the Vertex and Axisof Symmetry?

The x‐coordinate of the vertex isalso easily calculated from the quadratic formula because thevertex ishalfway between the two roots. If we average the two roots, the portion of the

formula disappearsand the resulting x‐value is 2 . The y‐value of the vertex must still

be calculated, but the x‐value can be read directly out of the quadratic formula.

Also, once the x‐value of the vertex isknown, the equation for the axisof symmetry isalso

known. It is the vertical line containing the vertex: 2 .

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