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Chapter 6

Review of Factoring and

Algebraic Fractions

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Section 6.2: Factoring: Common Factors and Difference of Squares

Factoring is the reverse of multiplying.

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

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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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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

1) 196 2) 4 25

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 41) 324 4 2) 16 81c cd m n

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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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I. 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

Section 6.3: Factoring Trinomials

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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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A General Strategy for Factoring Polynomials

Before 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 difference of two squares: x2 - y2 = (x + y)(x - y)

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

4. If an expression has four 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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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

Section 6.5: 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,

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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

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

Wrong! So very wrong!!

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The correct way to simplify the rational expression

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.,

Simplest form.

2

2

9

8 15

x

x x

3 3

3 5

x x

x x

3 3

3 5

x x

x x

3

5

x

x

3 and 5x x

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Simplify the rational expression

25

302

2

x

xx

25

302

2

x

xx

1. FACTOR the numerator and the denominator.2. Divide out any common FACTORS.

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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