factoring polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Chapter 13

Factoring Polynomials


13.1

The Greatest Common Factor

Martin-Gay, Developmental Mathematics, 2e 33



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.



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 Terms1. 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.

Greatest Common Factor



Find the GCF of each list of numbers.

a. 12 and 8 12 = 2 · 2 · 3

8 = 2 · 2 · 2So the GCF is 2 · 2 = 4.

b. 7 and 20 7 = 1 · 720 = 2 · 2 · 5There are no common prime factors so the GCF is 1.

Example



Find the GCF of each list of numbers.

a. 6, 8 and 46 6 = 2 · 3 8 = 2 · 2 · 246 = 2 · 23So the GCF is 2.

b. 144, 256 and 300144 = 2 · 2 · 2 · 2 · 3 · 3256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2300 = 2 · 2 · 3 · 5 · 5So the GCF is 2 · 2 = 4.

Example



a. x3 and x7

x3 = x · x · xx7 = 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

Find the GCF of each list of terms.

Example



Find the GCF of the following list of terms.

a3b2, a2b5 and a4b7

a3b2 = a · a · a · b · ba2b5 = 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

Example



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.

Helpful Hint

smallest exponent on xsmallest exponent on y



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.




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)

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)

Example



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.

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)

Factoring

Example:



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.



Example

Factor by grouping.

21x3y2 – 9x2y + 14xy – 6

= (21x3y2 – 9x2y) + (14xy – 6)= 3x2y(7xy – 3) + 2(7xy – 3)= (7xy – 3)(3x2 + 2)

factoring polynomials

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