Factoring Polynomials

1. Chapter 13 Factoring Polynomials

2. The Greatest Common Factor 13.1

3. Factoring Polynomials When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. The product is the factored form of the integer. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. The product is the factored form of the polynomial. The process of writing a polynomial as a product is called factoring the polynomial.

4. Greatest Common Factor Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. • Finding the GCF of a List of Integers or Terms • 1. Write each number or polynomial as a product of prime factors. • 2. Identify common prime factors. • 3. Take the product of all common prime factors. • If there are no common prime factors, GCF is 1.

5. Example Find the GCF of each list of numbers. • a. 12 and 8 12 = 2· 2· 3 8 = 2·2· 2 So the GCF is 2·2 = 4. • b. 7 and 20 7 = 1 · 7 20 = 2 · 2 · 5 There are no common prime factors so the GCF is 1.

6. Example Find the GCF of each list of numbers. • a. 6, 8 and 46 6 = 2 · 3 8 = 2· 2 · 2 46 = 2· 23 So the GCF is 2. • b. 144, 256 and 300 144 = 2 ·2 ·2 · 2 · 3 · 3 256 = 2·2 ·2 · 2 · 2 · 2 · 2 · 2 300 = 2·2 · 3 · 5 · 5 So the GCF is 2·2 = 4.

7. Example Find the GCF of each list of terms. a. x3 and x7 x3= x ·x·x x7 = x ·x·x·x ·x·x·x So the GCF is x · x· x = x3 b.6x5 and 4x3 6x5 = 2 · 3 · x · x· x 4x3 = 2 · 2 ·x ·x·x So the GCF is 2·x ·x·x = 2x3

8. Example Find the GCF of the following list of terms. • a3b2, a2b5 and a4b7 a3b2 = a ·a·a· b· b a2b5 = a · a· b· b · b· b· b a4b7 = a · a· a· a· b· b · b· b· b· b· b • So the GCF is a · a· b· b = a2b2

9. Helpful Hint Remember that the GCF of a list of terms contains the smallest exponent on each common variable. The GCF of x3y5, x6y4, and x4y6is x3y4. smallest exponent on x smallest exponent on y

10. Factoring Polynomials The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomial as a product by factoring out the GCF from all the terms. The remaining factors in each term will form a polynomial.

11. Example Factor out the GCF in each of the following polynomials. a. 6x3 – 9x2 + 12x = 3· x· 2 ·x2 – 3·x· 3 ·x + 3·x· 4 = 3x(2x2 – 3x + 4) b. 14x3y + 7x2y – 7xy = 7 ·x·y· 2 ·x2 + 7·x·y· x – 7·x·y· 1 = 7xy(2x2 + x – 1)

12. Example Factor out the GCF in each of the following polynomials. • a. 6(x + 2) – y(x + 2) = • 6 ·(x + 2) – y·(x + 2) = (x + 2)(6 – y) • b. xy(y + 1) – (y + 1) = xy·(y + 1) – 1 ·(y + 1) = (y + 1)(xy – 1)

13. Factoring Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial. This will usually be followed by additional steps in the process. Example: • Factor 90 + 15y2 – 18x – 3xy2. 90 + 15y2 – 18x – 3xy2 = 3(30 + 5y2 – 6x – xy2) = 3(5 · 6 + 5 · y2 – 6 · x – x · y2) = 3(5(6 + y2) – x (6 + y2)) = 3(6 + y2)(5 – x)

14. Factoring by Grouping Step 1: Group the terms in two groups so that each group has a common factor. Step 2: Factor out the GCF from each group. Step 3: If there is a common binomial factor, factor it out. Step 4: If not, rearrange the terms and try these steps again.

15. Example Factor by grouping. 21x3y2 – 9x2y + 14xy – 6 = (21x3y2 – 9x2y) + (14xy – 6) = 3x2y(7xy – 3) + 2(7xy – 3) = (7xy – 3)(3x2 + 2)