Sections 7.1, 7.2:

CHAPTER 7: POLYNOMIALS

Sections 7.1, 7.2: Sums, differences, products of polynomials

Quiz resultsQuiz results

A 73% hi h 100%Average 73%: high score 100%Problems:

Keeping track of negative signs— x — = +

— ÷ — = +

Function notationf(x) y: the result of the inputf(x) ~ y: the result of the inputx is the input

Retakes can be done up to 11 AM Oct 11Retakes can be done up to 11 AM Oct. 11

SumSum

Th lt f ddiThe result of addingThe sum of two positive values is a

i ipositiveThe sum of two negative values is a negativeWhen the values are different signs

Find the differenceUse the sign of the one with the larger b l labsolute value

DifferenceDifference

S bt tiSubtractionNeeds to be done in order from left to i hright

Subtract a negative value is the same as adding a positiveSuggest you change signs of terms that are subtracted and use adding rules

ProductProduct

Th lt f lti li ti f f tThe result of multiplication of factorsProduct of two positives is positiveProduct of two negatives is positiveIf signs are opposite, the product is negative

Term Term

A t t i bl d t f A constant, a variable or a product of a constant and one or more variable factorsfactorsTerms are separated from one another by additionadditionSubtraction in an expression or equation

d t b i t t d ddi needs to be interpreted as adding a negative term: Ch it i d ddi lChange its sign and use adding rules.

MonomialMonomial

Si l tSingle termMay be product of a constant and one or

i blmore variablesVariables may be raised to powers (exponents)Absence of a numeric constant implies the constant is 1

PolynomialPolynomial

M i l f i l tMonomial or sum of monomial termsNamed by number of terms

Binomial: 2Trinomial: 2

Polynomial degreePolynomial degree

Th t t t i tThe greatest exponents in any termAdd up the values of the exponents in

h each term4x3y2+2x2y2+9xy3

h1st term: 5th degree2nd term: 4th degree3 d t 4th d3rd term: 4th degree

Fifth degree polynomial

Terms are made of factorsTerms are made of factors

N i f t C ffi i tNumeric factor: CoefficientVariable factors represented by letters

“Like Terms” Like Terms

H tl th i bl tHave exactly the same variable setCombine the coefficients (numeric f ) b ddi ifactors) by additionRecall that the term has a sign

PositiveNegativeCombine by addition rules

Quadratic expression or functionQuadratic expression or function

S d d l i lSecond degree polynomialf(x) = 2x2 + 4x – 3Recall f(x) is the function, equivalent to the outputx is the input: example x=3f(3) = 2(3)2 + 4(3) – 3 = 2(9) +12 – 3 = 27( ) ( ) ( ) ( )

Quadratic function30f(x)

Quadratic function

G h i ll d b l20

25

Graph is called a parabolaVertex: 10

15

this has minimum value

Axis of symmetry 0

5

6 4 2 0 2 4x

Some open downwardHave maximum value -10

-5

-6 -4 -2 0 2 4

Cubic function 80Cubic function

G h i ti

60

Graph is a serpentine shapeM h h i l

20

40

May have horizontal line that crosses graph in more than -20

0

-6 -4 -2 0 2 4 6

graph in more than one place -40

20

-80

-60

Adding functionsAdding functions

f( ) 2 2 4 3f(x) = 2x2 + 4x — 3g(x) = —3x2 + 9x — 7(f+g)(x) = 2x2 + 4x — 3 —3x2 + 9x — 7Add like terms: watch the signs!!Do it vertical, not on one line

2x2 + 4x — 32x 4x 3+ —x2 + 9x — 7

x2 +13x 10—x2 +13x — 10

Subtracting functionsSubtracting functions

f(x) 2x2 + 4x 3f(x) = 2x2 + 4x — 3g(x) = —3x2 + 9x — 7(f )( ) 2 2 4 3 ( 3 2 9 7)(f—g)(x) = 2x2 + 4x — 3 — (—3x2 + 9x — 7)Change every sign of function being subtracted!! Then add watching signssubtracted!! Then add, watching signsDo it vertical, not on one line

2x2 + 4x 32x2 + 4x — 3+ +3x2 — 9x + 7

5x2 5x + 45x2 —5x + 4

Multiplying monomials Multiplying monomials (finding products)

Multiply numeric coefficientsCombine exponents on like variablesCombine exponents on like variables

x2 · x3 = Write factors without exponents to ‘see’ what Write factors without exponents to see what the exponent meansx·x · x·x·x = x5

Multiplying polynomial by Multiplying polynomial by monomial

2(3 5) 2(8) 16 i ht?2(3+5) = 2(8) = 16, right?“Distribute” multiplication over addition2(3) +2(5) = 6 + 10 = 16, same thing2x(3x2 + 5x) = 6x3 + 10x2( )

same thing with variables, just now terms are not ‘like’ so you cannot combine them

Product of binomialsProduct of binomials

(2 3)(2 5) (5)(7) 35(2+3)(2+5)=(5)(7)=35Multiply second factor by each term of fi f h di ibfirst factor, then distribute(2+3)(2+5)=2(2+5)+3(2+5)=14+21=35With variables: (x+3)(x+5)=x(x+5)+3(x+5)=( ) ( )x2+5x+3x+15=Combine like terms: x2+8x+15Combine like terms: x +8x+15

Product of binomialsProduct of binomialsF.O.I.L. methodL b l t Label terms

Firsts, Lasts, Outsides, InsidesF L F L

(x+3)(x+5)O I I O

M l i l d li i d d i Multiply: draw line, write product, draw, write… Firsts x2

Outsides +5xInsides +3xLasts +15

C bi lik t 2 8 15Combine like terms: x2+8x+15

Products of higher degree Products of higher degree polynomials

B th di lBe very methodicalDraw line for the product of two termsWrite the product of those termsDraw another line for product of termsWrite the product of those two termsEtc: DO NOT DRAW ALL THE LINES AND GO Etc: DO NOT DRAW ALL THE LINES AND GO BACK TO FIND THE PRODUCTS!!YOU WILL GET LOST!!YOU WILL GET LOST!!