6.1 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 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.

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

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

Example 1

Find the greatest common factor 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

It is sometimes useful to represent a number as a product of factors, for example, 12 as $2 times 6$ or $3 times 4.$ In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as $3 x squared plus 15 x,$ and end with its factors, $3 x open parenthesis x plus 5 close parenthesis.$ 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 4

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

The GCF of $8 x cubed y, −10 x squared y squared, and 12 x y cubed$ is $2 x y.$


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

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

Example 5

Factor: $−4 a^{3} + 36 a^{2} - 8 a .$

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


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

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

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 8

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

ⓐ

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

ⓑ

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.

Key Concepts

How to find the greatest common factor (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.
Distributive Property: If a, b, and c are real numbers, then

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

The form on the left is used to multiply. The form on the right is used to factor.
How to factor the 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.
Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

$Noun:, 7 is a factor of 14; Verb:, factor 3 from 3 a plus 3$
How to factor by grouping.
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.

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$10 p cubed q, 12 p q squared$

$2 p q$

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

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

$6 m squared n cubed$

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

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

$2 a$

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

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

$5 x cubed y$

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

Factor the Greatest Common Factor from a Polynomial

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

$6 m plus 9$

$3 open parenthesis 2 m plus 3 close parenthesis$

$14 p plus 35$

$9 n minus 63$

$9 open parenthesis n minus 7 close parenthesis$

$45 b minus 18$

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

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

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

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

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

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

$8 y cubed plus 16 y squared$

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

$12 x cubed minus 10 x$

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

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

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

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

$3 x (8 x^{2} - 4 x + 5)$

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

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

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

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

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

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

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

$−2 x minus 4$

$−2 open parenthesis x plus 2 close parenthesis$

$−3 b plus 12$

$−2 x^{3} + 18 x^{2} - 8 x$

$−2 x (x^{2} - 9 x + 4)$

$−5 y^{3} + 35 y^{2} - 15 y$

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

$−4 p q (p^{2} + 3 p q - 4 q)$

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

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

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

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

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

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

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

Factor by Grouping

In the following exercises, factor by grouping.

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

$(b + 5) (a + 3)$

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

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

$(y + 5) (8 y + 1)$

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

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

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

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

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

$(u - 1) (u + 6)$

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

$9 p^{2} + 12 p - 15 p - 20$

$(3 p - 5) (3 p + 4)$

$16 q^{2} + 20 q - 28 q - 35$

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

$(n - 6) (m - 4)$

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

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

$(x - 7) (2 x - 5)$

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

Mixed Practice

In the following exercises, factor.

$−18 x y squared minus 27 x squared y$

$−9 x y open parenthesis 2 y plus 3 x close parenthesis$

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

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

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

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

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

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

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

Writing Exercises

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

Answers will vary.

How do you check result after factoring a polynomial?

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 $y to the power 10 ?$ 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 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. 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.


Factor the GCF, $open parenthesis y plus 7 close parenthesis.$
Check on your own by multiplying.