Section P5Factoring Polynomials

Common Factors

Factoring a polynomial containing the sum of monomials mean finding an equivalent expression that is a product. In this section we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called prime.

Example

Factor:

2 3

2

64 28

5 ( 1) 10 ( 1)

x x

xy z xy z

Factoring by Grouping

Sometimes all of the terms of a polynomial may notcontain a common factor. However, by a suitable grouping of terms it may be possible to factor. Thisis called factoring by grouping.

Example

Factor by Grouping:

3 24 5 20x x x

Example

Factor by Grouping:

3 22 8 7 28x x x

Factoring Trinomials

Factors of 8

8,1 4,2 -8,-1 -4,-2

Sum of Factors

9 6 -9 -6

Factor:2 6 8x x

x x

4 2

+ +

Choose either two positive or two negative factors since the sign in front of the 8 is positive.

Factor: 22 9 5x x 2 x x +-

Possible factorizations Sum of outside and inside products

2 5 1

2 5 1

2 1 5

2 1 5

x x

x x

x x

x x

3

3

9

9

x

x

x

x

1 5

Since the sign in front of the 5 is a negative, one factor will be positive and one will be negative.

Example

Factor: 2 9 14x x

Possible Factorizations

Sum of Inside and Outside products

Example Factor: 23 2 21x x Possible Factorizations Sum of Inside and

outside Products

Factoring the Difference of

Two Squares

4

2 2

2

16

4 4

2 2 4

x

x x

x x x

Repeated Factorization- Another example

Can the sum of two squares be factored?

Example

Factor Completely:

24 9x

Example

Factor Completely:

249 81x

Example

Factor Completely:

4 481y x

Factoring Perfect Square Trinomials

Example

Factor:2 12 36x x

Example

Factor:

216 72 81x x

Factoring the Sum and Difference of Two Cubes

Example

Factor:

38 27x

Example

Factor:

3 3 3a b d

Example

Factor:

3 3125 64x y

A Strategy for Factoring Polynomials

Example

Factor Completely:3 212 60 75x x x

Example

Factor Completely:

4 81x

Example

Factor Completely:

327 64x

Example

Factor Completely:38 125x

Example

Factor Completely:

3 2 25 25x x x

Example

Factor Completely:

2 29 36x y

Factoring Algebraic Expressions Containing Fractional and Negative

Exponents

Expressions with fractional and negative exponents are not polynomials, but they can be factored using similar techniques. Find the greatest common factor with the smallest exponent in the terms.

3 1

4 4

31

4

3

4

3

4

3 6 6

6 3 6

6 4 6

2 6 2 3

x x x

x x x

x x

x x

3 11

4 4

Example

Factor and simplify:

1 3

4 4y y

Example

Factor and simplify:

1 3

2 25 5x x

Example

Factor and simplify:

2 1

3 33 3x x x

(a)

(b)

(c)

(d)

28 32x Factor Completely:

28 4

8 2 2

8 2 2

8 2 2

x

x x

x x

x x

(a)

(b)

(c)

(d)

Factor Completely:

3 327x y

2 2

2 2

2 2

2 2

(9 )(3 )

3

3 9 3

3 9 3

x y x y

x y x xy y

x y x xy y

x y x xy y