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Chapter 6

Factoring and Algebraic Fractions

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Section 6.2 Factoring: Common Factors and

Difference of Squares

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Factoring is the reverse of multiplying.

A polynomial or a factor is called _________________ if it contains no factors other than 1 or -1.

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THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor

2 2 2 3

4 2 3 2 2

1) 3 42 2) 10 15

3) 28 4 4 4) 12 18

x a b ab

x y xy y m n mn

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Solving Formulas Involving Factoring

1) Solve 2 2 2 forA wl lh wh l

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Solving Formulas Involving Factoring

2) Solve (2 ) (2 1) fork y y k y

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Factoring the Difference of Perfect Squares

Recall: (7 3)(7 3)x x

Difference of Squares:

2 2 _______________________a b

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Factoring the Difference of Perfect Squares

2 2 2

4 2 2

11) 196 2) 25

4

3) 144 4) 9 16

p m n

x y a

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Factor Completely:

HINT: Always check for a GCF first!!

3 2 4 4

2

1) 324 4 2) 16 81

3) 5 1

c cd m n

x

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Factoring by Grouping(Consider grouping method if polynomial has 4 terms)

1) Always start by checking for a GCF of all 4 terms. After you factor out the GCF or if the polynomial does not have a GCF other than 1, check if the remaining 4-term polynomial can be factored by grouping.

2) Determine if you can pair up the terms in such a way that each pair has its own common factor.

3) If so, factor out the common factor from each pair.

4) If the resulting terms have a common binomial factor, factor it out.

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2

2 2

1)3 15 2 10

2) 8 32 88 352

m m mn n

wv v wv v

Factor Completely

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2 2 2 21 1 2 23) ( )p R p r p R p r fluid flow

Factor Completely

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Section 6.3 Factoring Trinomials

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Factoring Trinomials in the Form 2x bx c

Recall: 2

2

( 5)( 8) 8 5 40

3 40

x x x x x

x x

F LO + I

To factor a trinomial is to reverse the multiplication process (UnFOIL)

2 7 12a a a a

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1) Always factor out the GCF first, if possible.

2) Write terms in descending order.

Before you attempt to Un-FOIL

3) Set up the binomial factors like this: (x )(x )

4) List the factor pairs of the LAST term

*If the LAST term is POSITIVE, then the signs must be the same (both + or both -)

*If the LAST term is NEGATIVE, then the signs must be different (one + and one -).

5) Find the pair whose sum is equal to the MIDDLE term

6) Check by multiplying the binomials (FOIL)

Now we begin

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2 2 2

2 2 2

1) 14 32 2) 9 18

3) 4 77 4) 14 49

m m x xy y

a a r rt t

Factor Completely

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3 25) 3 21 24q q q

Factor Completely

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Factoring Trinomials in the Form 2ax bx c

The Trial & Check Method:

1) Always factor out the GCF first, if possible.

2) Write terms in descending order.

Before you attempt to Un-FOIL

3) Set up the binomial factors like this: ( x )( x )

4) List the factor pairs of the FIRST term

5) List the factor pairs of the LAST term

6) Sub in possible factor pairs and ‘try’ them by multiplying the binomials (FOIL) until you find the winning combination; that is when O+I =MIDDLE term.

Now we begin

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Factor completely

2 21) 5 31 28 2) 4 13 28x x x x

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Factor completely

2 23) 2 3 15 4) 12 20x x x x

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Factor completely

2 35) 12 21 9g g g

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Factor completely

2 26) 4 20 25 121a a b

A tricky one!

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Section 6.4 The Sum and Difference of Cubes

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The Sum and Difference of CubesLearn these formulas!!

3 3 2 2

3 3 2 2

x y x y x xy y

x y x y x xy y

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Factor Completely

31) 27m 62) 8 1a

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Factor Completely

3 43) 3 192x y y

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A General Strategy for Factoring PolynomialsBefore you begin to factor, make sure the terms are written in descendingorder of the exponents on one of the variables. Rearrange the terms, if necessary.

1. Factor out all common factors (GCF). If your leading term is negative, factor out -1.

2. If an expression has two terms, check for the following types of polynomials:a) The difference of two squares: x2 - y2 = (x + y)(x - y)b) The sum of two cubes: x3 + y3 = (x + y)(x2 - xy + y2)c) The difference of two cubes: x3 - y3 = (x - y)(x2 + xy + y2)

3. If an expression has three terms, attempt to factor it as a trinomial.

4. If an expression has four or more terms, try factoring by grouping.

5. Continue factoring until each individual factor is prime. You may need to use a factoring technique more than once.

6. Check the results by multiplying the factors back out.

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Section 6.5 Equivalent Fractions

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Equivalent Fractions

The value of a fraction is unchanged if BOTH numerator and denominator are multiplied or divided by the same non-zero number.

5 5

12 12

15

36

3

3

Equivalent fractions

618 18

24 2 6

3

4 4

Equivalent fractions

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An algebraic fraction is a ratio of two polynomials.

Some examples of algebraic fractions are:

xm

m

xy

yx 6and,

19

13,

5

223

2

Algebraic fractions are also called rational expressions.

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Simplifying Algebraic Fractions

1. Factor the numerator and the denominator.

2. Divide out (cancel) the common FACTORS of the numerator and the denominator.

A fraction is in its simplest form if the numerator and denominator have no common factors other than 1 or -1.

(We say that the numerator and denominator are relatively prime.)

We use terms like “reduce”, “simplify”, or “put into lowest terms”.

Two simple steps for simplifying algebraic fractions:

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Cancel only common factors.

Do NOT cancel terms.

WARNING:

2 2

2

9

8 15

x x

x x

1

93

2x18 15x

5

Example: NEVER EVER NEVER do this!!!!!!!

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Simplify the rational expression

25

302

2

x

xx

25

302

2

x

xx

55

65

xx

xx

5x

6

5

x

x

5x

5

6

x

x

Here is the plan:1. Factor the numerator and the denominator.2. Divide out any common factors.

Notice in this example, , because the value of the denominator would be 0.,

5,5 xx

Simplest form.

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A Special Case

The numerator and denominator are OPPOSITES.

3 1 31) 1

3 3

1 12) 1

b a a bb a

a b a b a b

4 4 1

3) 14 1 4 1

x x

x x

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Examples

Simplify each fraction.

27 211)

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a a

a

2

2

42)

2 3 2

x

x x

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Example Simplify each fraction.

2

2

3 13 103)

5 4

a a

a a

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Example Simplify each fraction.

3

2

5 404)

2 4

c

c c

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Section 6.6 Multiplication and Division of

Algebraic Fractions

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Multiplying Fractions

a c ac

b d bd

Numerical Fractions: 16 15 16 15

25 96 25 96

1. Completely factor the numerator and denominator of each fraction.

2. Divide out common factors. (CANCEL)

3. Multiply the numerators and denominators of the reduced fractions: a c ac

b d bd

To multiply algebraic fractions:

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Simplify the given expressions involving multiplication.

3 3 2

7 2

4 7 2 41) 2)3 32 3 2

a b x x

b a x x x

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Simplify the given expressions involving multiplication.

3 5 3 4

2 2 4 3

1 43) 4)

1 4

a a t t t t

a a t t t t

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Dividing Fractions

a c a d ad

b d b c bc

1. Invert the second fraction and multiply. (Multiply by the reciprocal of the divisor.)

2. Completely factor the numerator and denominator of each fraction.

3. Divide out common factors. (CANCEL)

4. Multiply the numerators and denominators of the reduced fractions:

a c ac

b d bd

To divide algebraic fractions:

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Simplify the given expressions involving division.

5 2 2

2 9 2

9 3 3 31) 2)8 16 4 4 12

x x x y y x

y y x y x

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Simplify the given expression involving division.

3 2

2 2

3 5 143)

9 4 21

y y y y

y y y

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Section 6.7 Addition and Subtraction of

Algebraic Fractions

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To add or subtract like fractions (fractions with the same denominator), we add/subtract the numerators and keep the denominators the same.

Example:

2 2 2

5 7 11

12 12 12

3 9 1x

a a a

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If the fractions do NOT have a common denominator, we will first write equivalent fractions using the Least Common Denominator.

5 11

6 21

Example:

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Perform the operations and simplify.

2 3

5 11)3 6x y xy

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Perform the operations and simplify.

3 3 72)

2 2

x y

x y y x

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Perform the operations and simplify.

6 23)5 10 4 8

x x

x x

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Perform the operations and simplify.

2

2 2

2 6 24)

9 1 2 3

y y y

y y y y

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Complex Fractions

A fraction contained within a fraction.

1. Write the numerator as a single fraction.

2. Write the denominator as a single fraction.

3. Multiply the numerator by the reciprocal of the denominator.

4. Simplify.

5

4

xxx

Example

To simplify a complex fraction:

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5

1)4

xxx

Simplify the complex fraction.

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1

2)1

1

xx

x

Simplify the complex fraction.

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Section 6.8 Equations Involving Algebraic Fractions

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To solve an equation involving fractions:

Multiply each term of the equation (BOTH SIDES) by the LCD to rid the equation of fractions.

Example7 2

8 5 20

x

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We will use the same strategy for algebraic fractions.

1. Factor each denominator to determine the LCD

2. Multiply each term of the equations by the LCD to eliminate the fractions.

3. Remove grouping symbols by distributing (watch out for negative signs)

4. Combine like terms on each side of the equation.

5. Solve for the variable.

6. Check solutions in the ORIGINAL EQUATION. (Check for extraneous solutions*.)

*If an apparent solution causes a denominator of the original equation to equal zero, we reject that answer.

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Solve the equation and check the results.

4 3 101)3 3y y

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Solve the equation and check the results.

2

2 2 1 12)

4 2 8 2

a

a a a a

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Solve the equation and check the results.

2

2 3 5 53)

3 4 3 1

x

x x x x