7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

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

This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.

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.

First we’ll find the GCF of two numbers.

We summarize the steps we use to find the GCF below.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example 2

Find the greatest common factor of $27 x cubed and 18 x to the power 4$ .

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 $27 x cubed$ and $18 x to the power 4$ is $9 x cubed.$

Example 3

Find the GCF of $4 x squared y, 6 x y cubed$ .

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 $4 x squared y$ and $6 x y cubed$ is $2 xy$ .

Example 4

Find the GCF of: $21 x cubed, 9 x squared, 15 x$ .

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 $21 x cubed$ , $9 x squared$ and $15 x$ is $3 x.$

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as $2 \cdot 6 or 3 \cdot 4),$ in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

$\begin{matrix} 2 (x + 7) & factors \\ 2 \cdot x + 2 \cdot 7 \\ 2 x + 14 & product \end{matrix}$

Now we will start with a product, like $2 x plus 14$ , and end with its factors, $2 open parenthesis x plus 7 close parenthesis$ . 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.”

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

Factor: $5 a plus 5$ .

Find the GCF of 5a and 5.

Rewrite each term as a product using the GCF.
Use the Distributive Property "in reverse" to factor the GCF.
Check by mulitplying the factors to get the orginal polynomial.
$5 open parenthesis a plus 1 close parenthesis$
$5 \cdot a + 5 \cdot 1$
$5 a plus 5 ✓$

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example 8

Factor: $4 y^{2} + 24 y + 28$ .

We start by finding the GCF of all three terms.

Find the GCF of $4 y squared$ , $24 y$ and 28.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying.
$4 (y^{2} + 6 y + 7)$
$4 \cdot y^{2} + 4 \cdot 6 y + 4 \cdot 7$
$4 y^{2} + 24 y + 28 ✓$

Example 10

Factor: $21 x^{3} - 9 x^{2} + 15 x$ .

In a previous example we found the GCF of $21 x cubed, 9 x squared, 15 x$ to be $3 x$ .


Rewrite each term using the GCF, 3x.
Factor the GCF.
Check.
$3 x (7 x^{2} - 3 x + 5)$
$3 x \cdot 7 x^{2} - 3 x \cdot 3 x + 3 x \cdot 5$
$21 x^{3} - 9 x^{2} + 15 x ✓$

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

Example 12

Factor: $−8 y minus 24$ .

When the leading coefficient is negative, the GCF will be negative.

Ignoring the signs of the terms, we first find the GCF of 8y and 24 is 8. Since the expression −8y − 24 has a negative leading coefficient, we use −8 as the GCF.
Rewrite each term using the GCF.
Factor the GCF.
Check.
$−8 open parenthesis y plus 3 close parenthesis$
$−8 \cdot y + (−8) \cdot 3$
$−8 y minus 24 ✓$

Example 13

Factor: $−6 a squared plus 36 a$ .

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

Since the leading coefficient is negative, the GCF is negative, −6a.
Rewrite each term using the GCF.
Factor the GCF.
Check.
$−6 a open parenthesis a minus 6 close parenthesis$
$−6 a \cdot a + (−6 a) (−6)$
$−6 a squared plus 36 a ✓$

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating 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.)

Key Concepts

Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. Multiply the factors as in Example 2.
Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
1. Find the GCF of all the terms of the polynomial.
2. Rewrite each term as a product using the GCF.
3. Use the ‘reverse’ Distributive Property to factor the expression.
4. Check by multiplying the factors as in Example 5.
Factor by Grouping: To factor a polynomial with 4 four or more terms
1. Group terms with common factors.
2. Factor out the common factor in each group.
3. Factor the common factor from the expression.
4. Check by multiplying the factors as in Example 15.

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

8, 18

24, 40

72, 162

150, 275

10a, 50

5b, 30

$3 x, 10 x squared$

$x$

$21 b squared, 14 b$

$8 w squared, 24 w cubed$

$8 w squared$

$30 x squared, 18 x cubed$

$10 p cubed q, 12 p q squared$

$2 p q$

$8 a squared b cubed, 10 a b squared$

$12 m squared n cubed, 30 m to the power 5 n cubed$

$6 m squared n cubed$

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

$10 a cubed, 12 a squared, 14 a$

$2 a$

$20 y cubed, 28 y squared, 40 y$

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

$5 x cubed$

$27 p squared, 45 p cubed, 9 p to the power 4$

Factor the Greatest Common Factor from a Polynomial

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

$4 x plus 20$

$4 open parenthesis x plus 5 close parenthesis$

$8 y plus 16$

$6 m plus 9$

$3 open parenthesis 2 m plus 3 close parenthesis$

$14 p plus 35$

$9 q plus 9$

$9 open parenthesis q plus 1 close parenthesis$

$7 r plus 7$

$8 m minus 8$

$8 open parenthesis m minus 1 close parenthesis$

$4 n minus 4$

$9 n minus 63$

$9 open parenthesis n minus 7 close parenthesis$

$45 b minus 18$

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

$3 (x^{2} + 2 x - 3)$

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

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

$2 (4 p^{2} + 2 p + 1)$

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

$8 y cubed plus 16 y squared$

$8 y squared open parenthesis y plus 2 close parenthesis$

$12 x cubed minus 10 x$

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

$5 x (x^{2} - 3 x + 4)$

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

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

$6 y^{2} (2 x + 3 x^{2} - 5 y)$

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

$−2 x minus 4$

$−2 open parenthesis x plus 2 close parenthesis$

$−3 b plus 12$

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

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

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

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

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

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

Factor by Grouping

In the following exercises, factor by grouping.

$x y + 2 y + 3 x + 6$

$(y + 3) (x + 2)$

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

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

$(u + 2) (v - 9)$

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

$b^{2} + 5 b - 4 b - 20$

$(b - 4) (b + 5)$

$m^{2} + 6 m - 12 m - 72$

$p^{2} + 4 p - 9 p - 36$

$(p - 9) (p + 4)$

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

Mixed Practice

In the following exercises, factor.

$−20 x minus 10$

$−10 open parenthesis 2 x plus 1 close parenthesis$

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

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

$(x^{2} + 2) (3 x - 7)$

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

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

$(x + y) (x + 5)$

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

Everyday Math

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression $w squared minus 6 w$ , where $w =$ width. Factor the greatest common factor from the polynomial.

$w open parenthesis w minus 6 close parenthesis$

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression $−16 t^{2} + 80 t + 4$ . Factor the greatest common factor from the polynomial.

Writing Exercises

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

Answers will vary.

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

Self Check

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

This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

ⓑ 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. Whom 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.

Find the GCF of 12x and 60.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying the factors.
$12 open parenthesis x minus 5 close parenthesis$
$12 \cdot x - 12 \cdot 5$
$12 x minus 60 ✓$

Find the GCF of $5 x cubed$ and $25 x squared.$

Rewrite each term.
Factor the GCF.
Check.
$5 x squared open parenthesis x minus 5 close parenthesis$
$5 x^{2} \cdot x - 5 x^{2} \cdot 5$
$5 x cubed minus 25 x squared ✓$

Find the GCF of $8 m cubed$ , $12 m squared n$ , $20 m n squared$ .

Rewrite each term.
Factor the GCF.
Check.
$4 m (2 m^{2} - 3 m n + 5 n^{2})$
$4 m \cdot 2 m^{2} - 4 m \cdot 3 m n + 4 m \cdot 5 n^{2}$
$8 m^{3} - 12 m^{2} n + 20 m n^{2} ✓$