5.4 Factoring

Greatest Common Factor,Difference of Two Squares, Grouping, and Trinomials

Factoring a polynomial means expressing it as a product of other polynomials.

Factoring polynomials with a common monomial factor

(using GCF).

**Always look for a GCF before using any other factoring

method.

Factoring Method #1

Steps:

1. Find the greatest common factor (GCF)• Look for the number GCF, then look for variables in

common. The lowest exponent of each variable in common will be part of the GCF.

2. Divide the polynomial by the GCF. The quotient is the other factor.3. Express the polynomial as the product of the quotient and GCF.

Ex: 6c3d 12c2d 2 3cd

GCF 3cdSTEP #1:

Step#2: Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

3cd(2c2 4cd 1)

The answer should look like this:

Ex: 6c3d 12c2d 2 3cd

Factor these on your own looking for a GCF.

1. 6x3 3x2 12x

2. 5x2 10x 35

3. 16x3y4z 8x2y2z3 12xy3z 2

3x 2x2 x 4

5 x2 2x 7

4xy2z 4x2y2 2xz 2 3yz

Factoring polynomials that are a difference of perfect squares.

Factoring Method #2

DIFFERENCE OF PERFECT SQUARES

• When factoring using a difference of squares, look for the following 3 things:

1. Only 2 terms2. Minus sign between them3. Both terms must be perfect squares• If all 3 of the above are true, write 2 ( ), one with a + sign

and one with a – sign ( + ) ( - ) The terms in each of the parentheses will be the square root of each term.

A “Difference of Perfect Squares” is a binomial (*for

2 terms only*) and it factors like this:a2 b2 (a b)(a b)

To factor, express each term as a square of a monomial then apply the rule... a2 b2 (a b)(a b)

Ex: x2 16 x2 42

(x 4)(x 4)

Here is another example:

1

49x2 81

1

7x

2

92 1

7x 9

1

7x 9

Try these on your own:

1. x 2 121

2. 9y2 169x2

3. x4 16

Be careful!

x 11 x 11

3y 13x 3y 13x

x 2 x 2 x2 4

Factoring Method #3: Factor By Grouping

FACTOR BY GROUPING

1. When polynomials contain four terms, it is sometimes easier to group terms in order to factor.

2. Your goal is to create a common factor.

3. You can also move terms around in the polynomial to create a common factor.

4. Practice makes it easier to recognize common factors.

Factoring By Grouping1.Group the first two terms and the last two terms by putting

parentheses around them. 2. Factor out the GCF from each

group so that both sets of parentheses contain the same factors.

3. Factor our GCF again and write the answer as the product of two binomials.

Step 1: Group

b3 3b2 4b 12Ex:

b3 3b2 4b 12 Step 2: Factor out GCF from each group

b2 b 3 4 b 3 Step 3: Factor out GCF again

b 3 b2 4

2x3 16x2 8x 64Ex2:2 x3 8x2 4x 32

2 x3 8x2 4x 32 2 x 2 x 8 4 x 8 2 x 8 x2 4 2 x 8 x 2 x 2

Factoring ChartThis chart will help you to determine which method of factoring to use.Type Number of Terms

1. GCF 2 or more

2. Diff. Of Squares 2

3. Trinomials 3

4. Grouping 4

Box Method1. Make a box with four squares2. Make sure that the terms of the trinomial are

in descending order.3. Put the first term in the top left box. 4. Put the last term in the bottom right box. 5. Multiply those two terms together. 6. List factors of the product in #5 that will add

together to get the middle term.7. Put those in the other two boxes. 8. Find the GCF of each row and column – that

is your trinomial factored. Only take out a negative if the first box in the row or column is a negative number.

Trinomials

672 xx

Trinomials

132 2 aa

Trinomials

6136 2 cc

Trinomials

612 2 mm

1) Factor 2x2 + 9x + 10

(x + 2)(2x + 5)

2) Factor 6y2 - 13y - 5

(2y - 5)(3y + 1)

3) 12x2 + 11x - 5

(4x + 5)(3x - 1)

4) 5x - 6 + x2

x2 + 5x - 6

(x - 1)(x + 6)