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 Â· 3. However, 12 is not factored
completely as 4 Â· 3 because 4 is not a prime. To factor 12 completely, we write
12 = 2 Â· 2 Â· 3 (or 2^{2} Â· 3).
We use the distributive property to multiply a monomial and a binomial:
6x(2x  1) = 12x^{2}  6x
If we start with 12x^{2}  6x, we can use the distributive property to get
12x^{2}  6x = 6x(2x  1).
We have factored out 6x, which is a common factor of 12x^{2} and
6x. We could
have factored out just 3 to get
12x^{2}  6x = 3(4x^{2}  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) 8x^{2}y, 20xy^{3 }
b) 30a^{2}, 45a^{3 }b^{2}, 75a^{4}b
Solution
a) First factor each term completely:
8x^{2}y 20xy^{3} 
= 2^{3}x^{2}y = 2^{2} Â· 5xy^{3} 
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 2^{2}xy
or 4xy.
b) First factor each term completely:
30a^{2} 45a^{3}b^{2}
75a^{4}b 
= 2 Â· 3 Â· 5a^{2} 32 Â· 5a^{3}b^{2}
3 Â· 5^{2}a^{4}b 
The GCF is 3 Â· 5a^{2} or 15a^{2}.
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) 5x^{4}  10x^{3} + 15x^{2}
b) 8xy^{2} + 20x^{2}y
c) 60x^{5} + 24x^{3} + 36x^{2}
Solution
a) First factor each term completely:
5x^{4} = 5x^{4}, 10x^{3} = 2 Â· 5x^{3}, 15x^{2}
= 3 Â· 5x^{2}.
The GCF of the three terms is 5x^{2}. Now factor 5x^{2} out of each term:
5x^{4}  10x^{3} + 15x^{2} = 5x^{2}(x^{2}
 2x + 3)
b) The GCF for 8xy^{2} and 20x^{2}y is 4xy:
8xy^{2} + 20x^{2}y = 4xy(2y + 5x)
c) First factor each coefficient in 60x^{5} + 24x^{3} + 36x^{2}:
60 = 2^{2} Â· 3 Â· 5, 24 = 2^{3} Â· 3, 36 = 2^{2} Â· 3^{2}.
The GCF of the three terms is 2^{2} Â· 3x^{2} or 12x^{2}:
60x^{5} + 24x^{3} + 36x^{2} = 12x^{2}(5x^{3}
+ 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)
