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Factoring Polynomials and
Solving Equations
by Factoring
Copyright © Cengage Learning. All rights reserved.
5
Section
5.1
Factoring Out the Greatest Common
Factor; Factoring by Grouping
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Identify the greatest common factor of two or
more monomials.
2 Factor a polynomial containing a greatest
common factor.
3 Factor a polynomial containing a negative
greatest common factor.
4 Factor a polynomial containing a binomial
greatest common factor.
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Objectives
5 Factor a four-term polynomial using grouping.
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Factoring Out the Greatest Common Factor; Factoring
by Grouping
We will reverse the operation of multiplication and show
how to find the factors of a known product.
The process of finding the individual factors of a product is
called factoring.
We will limit our discussion of factoring polynomials to
those that factor using only rational numbers.
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1.
Identify the greatest common factor
of two or more monomials
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Identify the greatest common factor of two or more monomials
Recall that a natural number greater than 1 whose only
factors are 1 and the number itself is called a prime
number.
The prime numbers less than 50 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47
A natural number is said to be in prime-factored form if
it is written as the product of factors that are prime
numbers.
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Identify the greatest common factor of two or more monomials
To find the prime-factored form of a natural number, we
can use a factoring tree. For example, to find the
prime-factored form of 60, we proceed as follows:
Solution 1
1. Start with 60.
2. Factor 60 as 6  10.
3. Factor 6 and 10.
Solution 2
1. Start with 60.
2. Factor 60 as 4  15.
3. Factor 4 and 15.
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Identify the greatest common factor of two or more monomials
We stop when only prime numbers appear. In either case,
the prime factorization of 60 are 2  2  3  5.
Thus, the prime-factored form of 60 is 22  3  5.
This illustrates the fundamental theorem of arithmetic,
which states that there is exactly one prime factorization for
any natural number greater than 1.
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Greatest Common Factor of 2 Numbers
Greatest Common Factor (GCF) of 2 Numbers:
The largest natural number that divides each of these
numbers
E.g.,
The GCF of 42 and 60 is 6, because 6 is the largest
natural number that divides each of these numbers:
42
60
---- = 7 and ---- = 10
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6
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Identify the greatest common factor of two or more monomials
42 = 2  3  7
60 = 2  2  3  5
Algebraic monomials also can have a greatest common
factor. The right sides of the equations show the prime
factorizations of 6a2b3, 4a3b2, and 18a2b.
6a2b3 = 2  3  a  a  b  b  b
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Identify the greatest common factor of two or more monomials
4a3b2 = 2  2  a  a  a  b  b
18a2b = 2  3  3  a  a  b
Since all three of these monomials have one factor of 2,
two factors of a, and one factor of b, the GCF is
2aab
or
2a2b
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Example
Find the GCF of 10x3y2, 60x2y, and 30xy2.
Solution:
Find the prime factorization of each of the three monomial.
10x3y2 = 2  5  x  x  x  y  y
60x2y = 2  2  3  5  x  x  y
30xy2 = 2  3  5  x  y  y
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Example – Solution
cont’d
List each common factor the least number of times it
appears in any one monomial: 2, 5, x, and y.
Find the product of the factors in the list:
2  5  x  y = 10xy
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2.
Factor a polynomial containing a
greatest common factor
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Factor a polynomial containing a greatest common factor
Recall that the distributive property provides a way to
multiply a polynomial by a monomial. For example,
To reverse this process and factor the product 6x3 – 9x2y,
we can find the GCF of each term (which is 3x2) and then
use the distributive property.
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Factor a polynomial containing a greatest common factor
This process is called factoring out the greatest common
factor.
Finding the Greatest Common Factor (GCF)
1. Identify the number of terms.
2. Find the prime factorization of each term.
3. List each common factor the least number of times it
appears in any one term.
4. Find the product of the factors found in the list to obtain
the GCF.
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Example
Factor: 12y2 + 20y
Solution:
To find the GCF, we find the prime factorization of 12y2 and
20y.
12y2 = 2  2  3  y  y
20y = 2  2  5  y
GCF = 4y
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Example – Solution
cont’d
We can use the distributive property to factor out the GCF
of 4y.
12y2 + 20y = 4y  3y + 4y  5
= 4y(3y + 5)
Check by verifying that 4y(3y + 5) = 12y2 + 20y.
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3.
Factor a polynomial containing a
negative greatest common factor
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Factor a polynomial containing a negative greatest common factor
It is often useful to factor –1 out of a polynomial, especially
if the leading coefficient is negative.
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Example 6
Factor –1 out of –a3 + 2a2 – 4.
Solution:
–a3 + 2a2 – 4 = (–1)a3 + (–1)(–2a2) + (–1)4
= –1(a3 – 2a2 + 4)
Write each term
with a factor of –1.
Factor out the GCF,
–1.
= –(a3 – 2a2 + 4)
Check:
–(a3 – 2a2 + 4) = –a3 + 2a2 – 4
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4.
Factor a polynomial containing a
binomial greatest common factor
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Factor a polynomial containing a binomial greatest common factor
If the GCF of several terms is a polynomial, we can factor
out the common polynomial factor.
For example, since a + b is a common factor of (a + b)x
and (a + b)y, we can factor out the (a + b).
(a + b)x + (a + b)y = (a + b)(x + y)
We can check by verifying that
(a + b)(x + y) = (a + b)x + (a + b)y.
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Example
Factor a + 3 out of (a + 3) + (a + 3)2.
Solution:
Recall that a + 3 is equal to (a + 3)1 and that (a + 3)2 is
equal to (a + 3)(a + 3).
We can factor out a + 3 and simplify.
(a + 3) + (a + 3)2 = (a + 3)  1 + (a + 3)  (a + 3)
= (a + 3)[1 + (a + 3)]
Factor out
a + 3, the GCF.
= (a + 3)(a + 4)
Combine like terms.
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5.
Factor a four-term polynomial
using grouping
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Factor a four-term polynomial using grouping
Suppose we want to factor
ax + ay + cx + cy
Although no factor is common to all four terms, there is a
common factor of a in ax + ay and a common factor of c in
cx + cy.
In this case, we group the first two terms and group the last
two terms. We can factor out the a from the first two terms
and the c from the last two terms to obtain
ax + ay + cx + cy = a(x + y) + c(x + y)
= (x + y)(a + c)
Factor out (x + y).
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Factor a four-term polynomial using grouping
We can check the result by multiplication.
(x + y)(a + c) = ax + cx + ay + cy
= ax + ay + cx + cy
Thus, ax + ay + cx + cy factors as (x + y)(a + c). This type
of factoring is called factoring by grouping.
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Example
Factor: 2c + 2d – cd – d 2.
Solution:
2c + 2d – cd – d 2 = 2(c + d) – d(c + d)
= (c + d)(2 – d)
Factor out 2 from (2c + 2d)
and –d from (–cd – d2).
Factor out (c + d).
Check:
(c + d)(2 – d) = 2c – cd + 2d – d 2
= 2c + 2d – cd – d 2
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