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Factoring Out the Greatest Common Factor (GCF)

A polynomial may be factored by using division: If we know one factor of a polynomial, then we can use it as a divisor to obtain the other factor, the quotient. However, this technique is not very practical because the division process can be somewhat tedious, and it is not easy to obtain a factor to use as the divisor. In this section and the next two sections we will develop better techniques for factoring polynomials. These techniques will be used for solving equations and problems in the last section of this chapter.

A natural number larger than 1 that has no factors other than itself and 1 is called a prime number. The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 are the first nine prime numbers.

There are infinitely many prime numbers. To factor a natural number completely means to write it as a product of prime numbers. In factoring 12 we might write 12 = 4 · 3. However, 12 is not factored completely as 4 · 3 because 4 is not a prime. To factor 12 completely, we write 12 = 2 · 2 · 3 (or 22 · 3).

We use the distributive property to multiply a monomial and a binomial:

6x(2x - 1) = 12x2 - 6x

If we start with 12x2 - 6x, we can use the distributive property to get

12x2 - 6x = 6x(2x - 1).

We have factored out 6x, which is a common factor of 12x2 and -6x. We could have factored out just 3 to get

12x2 - 6x = 3(4x2 - 2x),

but this would not be factoring out the greatest common factor. The greatest common factor (GCF) is a monomial that includes every number or variable that is a factor of all of the terms of the polynomial.

 

We can use the following strategy for finding the greatest common factor of a group of terms.

 

Strategy for Finding the Greatest Common Factor (GCF)

1. Factor each term completely.

2. Write a product using each factor that is common to all of the terms.

3. On each of these factors, use an exponent equal to the smallest exponent that appears on that factor in any of the terms.

 

Example 1

The greatest common factor

Find the greatest common factor (GCF) for each group of terms.

a) 8x2y, 20xy3

b) 30a2, 45a3 b2, 75a4b

Solution

a) First factor each term completely:

8x2y

20xy3

= 23x2y

= 22 · 5xy3

The factors common to both terms are 2, x, and y. In the GCF we use the smallest exponent that appears on each factor in either of the terms. So the GCF is 22xy or 4xy.

b) First factor each term completely:

30a2

45a3b2

75a4b

= 2 · 3 · 5a2

32 · 5a3b2

3 · 52a4b

The GCF is 3 · 5a2 or 15a2.

To factor out the GCF from a polynomial, find the GCF for the terms, then use the distributive property to factor it out.

 

Example 2

Factoring out the greatest common factor

Factor each polynomial by factoring out the GCF.

a) 5x4 - 10x3 + 15x2

b) 8xy2 + 20x2y

c) 60x5 + 24x3 + 36x2

Solution

a) First factor each term completely:

5x4 = 5x4, 10x3 = 2 · 5x3, 15x2 = 3 · 5x2.

The GCF of the three terms is 5x2. Now factor 5x2 out of each term:

5x4 - 10x3 + 15x2 = 5x2(x2 - 2x + 3)

b) The GCF for 8xy2 and 20x2y is 4xy:

8xy2 + 20x2y = 4xy(2y + 5x)

c) First factor each coefficient in 60x5 + 24x3 + 36x2:

60 = 22 · 3 · 5, 24 = 23 · 3, 36 = 22 · 32.

The GCF of the three terms is 22 · 3x2 or 12x2:

60x5 + 24x3 + 36x2 = 12x2(5x3 + 2x + 3)

In the next example the common factor in each term is a binomial.

 

Example 3

Factoring out a binomial

Factor.

a) (x + 3)w + (x + 3)a

b) x(x - 9) - 4(x - 9)

Solution

a) We treat x + 3 like a common monomial when factoring:

(x + 3)w + (x + 3)a = (x + 3)(w + a)

b) Factor out the common binomial x - 9:

x(x - 9) - 4(x - 9) = (x - 4)(x - 9)

 
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