OpenStax-CNX module: m18870
1
Factoring Polynomials: Finding the
factors of a Monomial
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 2.0†
Abstract
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an
essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken
great care in developing the student's understanding of the factorization process. The technique is
consistently illustrated by displaying an empty set of parentheses and describing the thought process
used to discover the terms that are to be placed inside the parentheses. The factoring scheme for special
products is presented with both verbal and symbolic descriptions, since not all students can interpret
symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect
and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with
leading coecients dierent from 1. Objectives of this module: be reminded of products of polynomials,
be able to determine a second factor of a polynomial given a rst factor.
1 Overview
ˆ Products of Polynomials
ˆ Factoring
2 Products of Polynomials
Previously, we studied multiplication of polynomials (Section ). We were given factors and asked to nd
their product, as shown below.
Example 1
Given the factors 4and 8, nd the product. 4 · 8 = 32. The product is 32.
Example 2
Given the factors 6x2 and 2x − 7, nd the product.
6x2 (2x − 7) = 12x3 − 42x2
The product is 12x3 − 42x2 .
Example 3
Given the factors x − 2y and 3x + y , nd the product.
∗
†
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OpenStax-CNX module: m18870
(x − 2y) (3x + y)
2
3x2 + xy − 6xy − 2y 2
=
= 3x2 − 5xy − 2y 2
The product is 3x − 5xy − 2y 2 .
2
Example 4
Given the factors a + 8 and a + 8, nd the product.
2
(a + 8) = a2 + 16a + 64
The product is a2 + 16a + 64.
3 Factoring
Now, let's reverse the situation. We will be given the product, and we will try to nd the factors. This
process, which is the reverse of multiplication, is called factoring.
Factoring
Factoring
is the process of determining the factors of a given product.
4 Sample Set A
Example 5
The number 24 is the product, and one factor is 6. What is the other factor?
We're looking for a number
such that 6 ·
= 24. We know from experience that
= 4. As problems become progressively more complex, our experience may not give us the
solution directly. We need amethod
for nding factors. To develop this method we can use the
relatively simple problem 6 ·
= 24 as a guide.
To nd the number
24
6
, we would divide 24 by 6.
=4
The other factor is 4.
Example 6
The product is 18x3 y 4 z 2 and one factor is 9xy 2 z . What is the other factor?
We know that since 9xy 2 z is a factor of 18x3 y 4 z 2 , there must be some quantity
that 9xy 2 z ·
18x3 y 4 z 2
9xy 2 z
such
= 18x3 y 4 z 2 . Dividing 18x3 y 4 z 2 by 9xy 2 z , we get
= 2x2 y 2 z
Thus, the other factor is 2x2 y 2 z .
Checking will convince us that 2x2 y 2 z is indeed the proper factor.
2x2 y 2 z
9xy 2 z
=
18x2+1 y 2+2 z 1+1
=
18x3 y 4 z 2
We should try to nd the quotient mentally and avoid actually writing the division problem.
Example 7
The product is −21a5 bn and 3ab4 is a factor. Find the other factor.
Mentally dividing −21a5 bn by 3ab4 , we get
−21a5 bn
3ab4
= −7a5−1 bn−4 = −7a4 bn−4
Thus, the other factor is −7a4 bn−4 .
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OpenStax-CNX module: m18870
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5 Practice Set A
Exercise 1
(Solution on p. 6.)
Exercise 2
(Solution on p. 6.)
The product is 84 and one factor is 6. What is the other factor?
The product is 14x3 y 2 z 5 and one factor is 7xyz . What is the other factor?
6 Exercises
In the following problems, the rst quantity represents the product and the second quantity represents a
factor of that product. Find the other factor.
Exercise 3
(Solution on p. 6.)
30, 6
Exercise 4
45, 9
Exercise 5
(Solution on p. 6.)
10a, 5
Exercise 6
16a, 8
Exercise 7
(Solution on p. 6.)
21b, 7b
Exercise 8
15a, 5a
Exercise 9
(Solution on p. 6.)
20x3 , 4
Exercise 10
30y 4 , 6
Exercise 11
(Solution on p. 6.)
8x4 , 4x
Exercise 12
16y 5 , 2y
Exercise 13
(Solution on p. 6.)
6x2 y, 3x
Exercise 14
9a4 b5 , 9a4
Exercise 15
(Solution on p. 6.)
15x2 b4 c7 , 5x2 bc6
Exercise 16
25a3 b2 c, 5ac
Exercise 17
18x2 b5 , − 2xb4
Exercise 18
22b8 c6 d3 , − 11b8 c4
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(Solution on p. 6.)
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Exercise 19
(Solution on p. 6.)
−60x5 b3 f 9 , − 15x2 b2 f 2
Exercise 20
39x4 y 5 z 11 , 3xy 3 z 10
Exercise 21
(Solution on p. 6.)
147a20 b6 c18 d2 , 21a3 bd
Exercise 22
−121a6 b8 c10 , 11b2 c5
Exercise 23
1 4 3
8x y ,
(Solution on p. 6.)
1
3
2 xy
Exercise 24
7x2 y 3 z 2 , 7x2 y 3 z
Exercise 25
(Solution on p. 6.)
5a4 b7 c3 d2 , 5a4 b7 c3 d
Exercise 26
14x4 y 3 z 7 , 14x4 y 3 z 7
Exercise 27
(Solution on p. 6.)
12a3 b2 c8 , 12a3 b2 c8
Exercise 28
2
6(a + 1) (a + 5) , 3(a + 1)
2
Exercise 29
(Solution on p. 6.)
3
8(x + y) (x − 2y) , 2 (x − 2y)
Exercise 30
6
2
2
14(a − 3) (a + 4) , 2(a − 3) (a + 4)
Exercise 31
10
12
7
7
(Solution on p. 6.)
26(x − 5y) (x − 3y) , − 2(x − 5y) (x − 3y)
Exercise 32
4
8
4
2
34(1 − a) (1 + a) , − 17(1 − a) (1 + a)
Exercise 33
(Solution on p. 6.)
(x + y) (x − y) , x − y
Exercise 34
(a + 3) (a − 3) , a − 3
Exercise 35
48xn+3 y 2n−1 , 8x3 y n+5
Exercise 36
0.0024x4n y 3n+5 z 2 , 0.03x3n y 5
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(Solution on p. 6.)
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7 Exercises for Review
Exercise 37
()
(Solution on p. 6.)
3
Simplify x4 y 0 z 2 .
Exercise 38
() Simplify −{− [− (−|6|)]}.
Exercise 39
2
() Find the product. (2x − 4) .
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(Solution on p. 6.)
OpenStax-CNX module: m18870
Solutions to Exercises in this Module
Solution to Exercise (p. 3)
14
Solution to Exercise (p. 3)
2x2 yz 4
Solution to Exercise (p. 3)
5
Solution to Exercise (p. 3)
2a
Solution to Exercise (p. 3)
3
Solution to Exercise (p. 3)
5x3
Solution to Exercise (p. 3)
2x3
Solution to Exercise (p. 3)
2xy
Solution to Exercise (p. 3)
3b3 c
Solution to Exercise (p. 3)
−9xb
Solution to Exercise (p. 3)
4x3 bf 7
Solution to Exercise (p. 4)
7a17 b5 c18 d
Solution to Exercise (p. 4)
1 3
4x
Solution to Exercise (p. 4)
d
Solution to Exercise (p. 4)
1
Solution to Exercise (p. 4)
3
4(x + y)
Solution to Exercise (p. 4)
3
−13(x − 5y) (x − 3y)
5
Solution to Exercise (p. 4)
(x + y)
Solution to Exercise (p. 4)
6xn y n−6
Solution to Exercise (p. 5)
x12 z 6
Solution to Exercise (p. 5)
4x2 − 16x + 16
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