Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

Greatest Common Factor and Factor by Grouping

Learning Objectives

By the end of this section, you will be able to:

Find the greatest common factor of two or more expressions
Factor the greatest common factor from a polynomial
Factor by grouping

Be Prepared 6.1

Before you get started, take this readiness quiz.

Factor 56 into primes.
If you missed this problem, review Example 1.2.
Find the least common multiple (LCM) of 18 and 24.
If you missed this problem, review Example 1.3.
Multiply: $−3 a (7 a + 8 b) . −3 a (7 a + 8 b) .$
If you missed this problem, review Example 5.26.

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

8 times 7 is 56. Here 8 and 7 are factors and 56 is the product. An arrow pointing from 8 times 7 to 56 is labeled multiply. An arrow pointing from 56 to 8 times 7 is labeled factor. 2x open parentheses x plus 3 close parentheses equals 2x squared plus 6x. Here the left side of the equation is labeled factors and the right side is labeled products.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

We summarize the steps we use to find the greatest common factor.

How To

Find the greatest common factor (GCF) of two expressions.

Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
Step 3. Bring down the common factors that all expressions share.
Step 4. Multiply the factors.

The next example will show us the steps to find the greatest common factor of three expressions.

Example 6.1

Find the greatest common factor of $21 x^{3}, 9 x^{2}, 15 x . 21 x^{3}, 9 x^{2}, 15 x .$

Show solution

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors.
Multiply the factors.
	The GCF of $21 x^{3} 21 x^{3}$ , $9 x^{2} 9 x^{2}$ and $15 x 15 x$ is $3 x 3 x$ .

Try It 6.1

Find the greatest common factor: $25 m^{4}, 35 m^{3}, 20 m^{2} . 25 m^{4}, 35 m^{3}, 20 m^{2} .$

Try It 6.2

Find the greatest common factor: $14 x^{3}, 70 x^{2}, 105 x . 14 x^{3}, 70 x^{2}, 105 x .$

Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as $2 \cdot 6 2 \cdot 6$ or $3 \cdot 4 . 3 \cdot 4 .$ In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as $3 x^{2} + 15 x, 3 x^{2} + 15 x,$ and end with its factors, $3 x (x + 5) . 3 x (x + 5) .$ To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive Property

If a, b, and c are real numbers, then

a (b + c) = a b + a c and a b + a c = a (b + c) a (b + c) = a b + a c

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 6.2

How to Use the Distributive Property to factor a polynomial

Factor: $8 m^{3} - 12 m^{2} n + 20 m n^{2} . 8 m^{3} - 12 m^{2} n + 20 m n^{2} .$

Show solution

Solution

Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.

In step 3, use the reverse Distributive Property to factor the expression as 4m open parentheses 2 m squared minus 3 mn plus 5 n squared close parentheses.

Step 4 is to check by multiplying the factors. By multiplying the factors, we get the original polynomial.

Try It 6.3

Factor: $9 x y^{2} + 6 x^{2} y^{2} + 21 y^{3} . 9 x y^{2} + 6 x^{2} y^{2} + 21 y^{3} .$

Try It 6.4

Factor: $3 p^{3} - 6 p^{2} q + 9 p q^{3} . 3 p^{3} - 6 p^{2} q + 9 p q^{3} .$

How To

Factor the greatest common factor from a polynomial.

Step 1. Find the GCF of all the terms of the polynomial.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the “reverse” Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb:

\begin{matrix} Noun: & 7 is a factor of 14 \\ Verb: & factor 3 from 3 a + 3 \end{matrix} \begin{matrix} Noun: \\ Verb: \end{matrix}

Example 6.3

Factor: $5 x^{3} - 25 x^{2} . 5 x^{3} - 25 x^{2} .$

Show solution

Solution

Find the GCF of $5 x^{3} 5 x^{3}$ and $25 x^{2} . 25 x^{2} .$


Rewrite each term.
Factor the GCF.
Check: $\begin{matrix} 5 x^{2} (x - 5) \\ 5 x^{2} \cdot x - 5 x^{2} \cdot 5 \\ 5 x^{3} - 25 x^{2} ✓ \end{matrix}$

Try It 6.5

Factor: $2 x^{3} + 12 x^{2} . 2 x^{3} + 12 x^{2} .$

Try It 6.6

Factor: $6 y^{3} - 15 y^{2} . 6 y^{3} - 15 y^{2} .$

Example 6.4

Factor: $8 x^{3} y - 10 x^{2} y^{2} + 12 x y^{3} . 8 x^{3} y - 10 x^{2} y^{2} + 12 x y^{3} .$

Show solution

Solution

The GCF of $8 x^{3} y, −10 x^{2} y^{2}, and 12 x y^{3} 8 x^{3} y, −10 x^{2} y^{2},$ is $2 x y . 2 x y .$


Rewrite each term using the GCF, $2 x y . 2 x y .$
Factor the GCF.
Check: $\begin{matrix} 2 x y (4 x^{2} - 5 x y + 6 y^{2}) \\ 2 x y \cdot 4 x^{2} - 2 x y \cdot 5 x y + 2 x y \cdot 6 y^{2} \\ 8 x^{3} y - 10 x^{2} y^{2} + 12 x y^{3} ✓ \end{matrix}$

Try It 6.7

Factor: $15 x^{3} y - 3 x^{2} y^{2} + 6 x y^{3} . 15 x^{3} y - 3 x^{2} y^{2} + 6 x y^{3} .$

Try It 6.8

Factor: $8 a^{3} b + 2 a^{2} b^{2} - 6 a b^{3} . 8 a^{3} b + 2 a^{2} b^{2} - 6 a b^{3} .$

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 6.5

Factor: $−4 a^{3} + 36 a^{2} - 8 a . −4 a^{3} + 36 a^{2} - 8 a .$

Show solution

Solution

The leading coefficient is negative, so the GCF will be negative.


Rewrite each term using the GCF, $−4 a . −4 a .$
Factor the GCF.
Check: $\begin{matrix} −4 a (a^{2} - 9 a + 2) \\ −4 a \cdot a^{2} - (−4 a) \cdot 9 a + (−4 a) \cdot 2 \\ −4 a^{3} + 36 a^{2} - 8 a ✓ \end{matrix}$

Try It 6.9

Factor: $−4 b^{3} + 16 b^{2} - 8 b . −4 b^{3} + 16 b^{2} - 8 b .$

Try It 6.10

Factor: $−7 a^{3} + 21 a^{2} - 14 a . −7 a^{3} + 21 a^{2} - 14 a .$

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 6.6

Factor: $3 y (y + 7) - 4 (y + 7) . 3 y (y + 7) - 4 (y + 7) .$

Show solution

Solution

The GCF is the binomial $y + 7 . y + 7 .$


Factor the GCF, $(y + 7) . (y + 7) .$
Check on your own by multiplying.

Try It 6.11

Factor: $4 m (m + 3) - 7 (m + 3) . 4 m (m + 3) - 7 (m + 3) .$

Try It 6.12

Factor: $8 n (n - 4) + 5 (n - 4) . 8 n (n - 4) + 5 (n - 4) .$

Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.

Example 6.7

How to Factor a Polynomial by Grouping

Factor by grouping: $x y + 3 y + 2 x + 6 . x y + 3 y + 2 x + 6 .$

Show solution

Solution

Step 1 is to group the terms with common factors. There is no greatest common factor in all the four terms of xy plus 3y plus 2x plus 6. So, separate the first two terms from the second two.

Step 2 is to factor out the common factor in each group. By factoring the GCF from the first 2 terms, we get y open parentheses x plus 3 close parentheses plus 2x plus 6. Factoring the GCF from the second 2 terms, we get y open parentheses x plus 3 close parentheses plus 2 open parentheses x plus 3 close parentheses.

Step 3 is to factor the common factor from the expression. Notice that each term has a common factor of x plus 3. By factoring this out, we get open parentheses x plus 3 close parentheses open parentheses y plus 2 close parentheses

Step 4 is to check by multiplying the expressions to get the result xy plus 3y plus 2x plus 6.

Try It 6.13

Factor by grouping: $x y + 8 y + 3 x + 24 . x y + 8 y + 3 x + 24 .$

Try It 6.14

Factor by grouping: $a b + 7 b + 8 a + 56 . a b + 7 b + 8 a + 56 .$

How To

Factor by grouping.

Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Example 6.8

Factor by grouping: ⓐ $x^{2} + 3 x - 2 x - 6 x^{2} + 3 x - 2 x - 6$ ⓑ $6 x^{2} - 3 x - 4 x + 2 . 6 x^{2} - 3 x - 4 x + 2 .$

Show solution

Solution

ⓐ
$\begin{matrix} There is no GCF in all four terms. & x^{2} + 3 x - 2 x - 6 \\ Separate into two parts. & x^{2} + 3 x −2 x - 6 \\ \begin{matrix} Factor the GCF from both parts. Be careful \\ with the signs when factoring the GCF from \\ the last two terms. \end{matrix} & x (x + 3) - 2 (x + 3) \\ Factor out the common factor. & (x + 3) (x - 2) \\ Check on your own by multiplying. \end{matrix} \begin{matrix} There is no GCF in all four terms. & x^{2} + 3 x - 2 x - 6 \\ Separate into two parts. & x^{2} + 3 x \\ \begin{matrix} Factor the GCF from both parts. Be careful \\ with the signs when factoring the GCF from \\ the last two terms. \end{matrix} & x (x + 3) - 2 (x + 3) \\ Factor out the common factor. & (x + 3) (x - 2) \\ Check on your own by multiplying. \end{matrix}$

ⓑ
$\begin{matrix} There is no GCF in all four terms. & 6 x^{2} - 3 x - 4 x + 2 \\ Separate into two parts. & 6 x^{2} - 3 x −4 x + 2 \\ Factor the GCF from both parts. & 3 x (2 x - 1) - 2 (2 x - 1) \\ Factor out the common factor. & (2 x - 1) (3 x - 2) \\ Check on your own by multiplying. \end{matrix} \begin{matrix} There is no GCF in all four terms. \\ Separate into two parts. \\ Factor the GCF from both parts. \\ Factor out the common factor. \\ Check on your own by multiplying. \end{matrix}$

Try It 6.15

Factor by grouping: ⓐ $x^{2} + 2 x - 5 x - 10 x^{2} + 2 x - 5 x - 10$ ⓑ $20 x^{2} - 16 x - 15 x + 12 . 20 x^{2} - 16 x - 15 x + 12 .$

Try It 6.16

Factor by grouping: ⓐ $y^{2} + 4 y - 7 y - 28 y^{2} + 4 y - 7 y - 28$ ⓑ $42 m^{2} - 18 m - 35 m + 15 . 42 m^{2} - 18 m - 35 m + 15 .$

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$10 p^{3} q, 12 p q^{2} 10 p^{3} q, 12 p q^{2}$

$8 a^{2} b^{3}, 10 a b^{2} 8 a^{2} b^{3}, 10 a b^{2}$

$12 m^{2} n^{3}, 30 m^{5} n^{3} 12 m^{2} n^{3}, 30 m^{5} n^{3}$

$28 x^{2} y^{4}, 42 x^{4} y^{4} 28 x^{2} y^{4}, 42 x^{4} y^{4}$

$10 a^{3}, 12 a^{2}, 14 a 10 a^{3}, 12 a^{2}, 14 a$

$20 y^{3}, 28 y^{2}, 40 y 20 y^{3}, 28 y^{2}, 40 y$

$35 x^{3} y^{2}, 10 x^{4} y, 5 x^{5} y^{3} 35 x^{3} y^{2}, 10 x^{4} y, 5 x^{5} y^{3}$

$27 p^{2} q^{3}, 45 p^{3} q^{4}, 9 p^{4} q^{3} 27 p^{2} q^{3}, 45 p^{3} q^{4}, 9 p^{4} q^{3}$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$6 m + 9 6 m + 9$

10.

$14 p + 35 14 p + 35$

11.

$9 n - 63 9 n - 63$

12.

$45 b - 18 45 b - 18$

13.

$3 x^{2} + 6 x - 9 3 x^{2} + 6 x - 9$

14.

$4 y^{2} + 8 y - 4 4 y^{2} + 8 y - 4$

15.

$8 p^{2} + 4 p + 2 8 p^{2} + 4 p + 2$

16.

$10 q^{2} + 14 q + 20 10 q^{2} + 14 q + 20$

17.

$8 y^{3} + 16 y^{2} 8 y^{3} + 16 y^{2}$

18.

$12 x^{3} - 10 x 12 x^{3} - 10 x$

19.

$5 x^{3} - 15 x^{2} + 20 x 5 x^{3} - 15 x^{2} + 20 x$

20.

$8 m^{2} - 40 m + 16 8 m^{2} - 40 m + 16$

21.

$24 x^{3} - 12 x^{2} + 15 x 24 x^{3} - 12 x^{2} + 15 x$

22.

$24 y^{3} - 18 y^{2} - 30 y 24 y^{3} - 18 y^{2} - 30 y$

23.

$12 x y^{2} + 18 x^{2} y^{2} - 30 y^{3} 12 x y^{2} + 18 x^{2} y^{2} - 30 y^{3}$

24.

$21 p q^{2} + 35 p^{2} q^{2} - 28 q^{3} 21 p q^{2} + 35 p^{2} q^{2} - 28 q^{3}$

25.

$20 x^{3} y - 4 x^{2} y^{2} + 12 x y^{3} 20 x^{3} y - 4 x^{2} y^{2} + 12 x y^{3}$

26.

$24 a^{3} b + 6 a^{2} b^{2} - 18 a b^{3} 24 a^{3} b + 6 a^{2} b^{2} - 18 a b^{3}$

27.

$−2 x - 4 −2 x - 4$

28.

$−3 b + 12 −3 b + 12$

29.

$−2 x^{3} + 18 x^{2} - 8 x −2 x^{3} + 18 x^{2} - 8 x$

30.

$−5 y^{3} + 35 y^{2} - 15 y −5 y^{3} + 35 y^{2} - 15 y$

31.

$−4 p^{3} q - 12 p^{2} q^{2} + 16 p q^{2} −4 p^{3} q - 12 p^{2} q^{2} + 16 p q^{2}$

32.

$−6 a^{3} b - 12 a^{2} b^{2} + 18 a b^{2} −6 a^{3} b - 12 a^{2} b^{2} + 18 a b^{2}$

33.

$5 x (x + 1) + 3 (x + 1) 5 x (x + 1) + 3 (x + 1)$

34.

$2 x (x - 1) + 9 (x - 1) 2 x (x - 1) +$

35.

$3 b (b - 2) - 13 (b - 2) 3 b (b - 2) - 13 (b - 2)$

36.

$6 m (m - 5) - 7 (m - 5) 6 m (m - 5) - 7 (m - 5)$

Factor by Grouping

In the following exercises, factor by grouping.

37.

$a b + 5 a + 3 b + 15 a b + 5 a + 3 b + 15$

38.

$c d + 6 c + 4 d + 24 c d + 6 c + 4 d + 24$

39.

$8 y^{2} + y + 40 y + 5 8 y^{2} + y + 40 y + 5$

40.

$6 y^{2} + 7 y + 24 y + 28 6 y^{2} + 7 y + 24 y + 28$

41.

$u v - 9 u + 2 v - 18 u v - 9 u + 2 v - 18$

42.

$p q - 10 p + 8 q - 80 p q - 10 p + 8 q - 80$

43.

$u^{2} - u + 6 u - 6 u^{2} - u + 6 u - 6$

44.

$x^{2} - x + 4 x - 4 x^{2} - x + 4 x - 4$

45.

$9 p^{2} - 3 p - 20 9 p^{2} - 3 p - 20$

46.

$16 q^{2} - 8 q - 35 16 q^{2} - 8 q - 35$

47.

$m n - 6 m - 4 n + 24 m n - 6 m - 4 n + 24$

48.

$r^{2} - 3 r - r + 3 r^{2} - 3 r - r + 3$

49.

$2 x^{2} - 14 x - 5 x + 35 2 x^{2} - 14 x - 5 x + 35$

50.

$4 x^{2} - 36 x - 3 x + 27 4 x^{2} - 36 x - 3 x + 27$

Mixed Practice

In the following exercises, factor.

51.

$−18 x y^{2} - 27 x^{2} y −18 x y^{2} - 27 x^{2} y$

52.

$−4 x^{3} y^{5} - x^{2} y^{3} + 12 x y^{4} −4 x^{3} y^{5} - x^{2} y^{3} + 12 x y^{4}$

53.

$3 x^{3} - 7 x^{2} + 6 x - 14 3 x^{3} - 7 x^{2} + 6 x - 14$

54.

$x^{3} + x^{2} - x - 1 x^{3} + x^{2} - x - 1$

55.

$x^{2} + x y + 5 x + 5 y x^{2} + x y + 5 x + 5 y$

56.

$5 x^{3} - 3 x^{2} + 5 x - 3 5 x^{3} - 3 x^{2} + 5 x - 3$

Writing Exercises

57.

What does it mean to say a polynomial is in factored form?

58.

How do you check result after factoring a polynomial?

59.

The greatest common factor of 36 and 60 is 12. Explain what this means.

60.

What is the GCF of $y^{4}, y^{5}, y^{4}, y^{5},$ and $y^{10} ? y^{10} ?$ Write a general rule that tells you how to find the GCF of $y^{a}, y^{b}, y^{a}, y^{b},$ and $y^{c} . y^{c} .$

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no I don’t get it. The first column has the following statements: find the greatest common factor of 2 or more expressions, factor the greatest common factor from a polynomial, factor by grouping. The remaining columns are blank.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.