Topics
5. Algebraic Expressions
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United States of America
Grade 7

5.04 Factor with integers

Lesson
Practice

Introduction

Previously, we looked at how to use the distributive property to remove parentheses from algebraic expressions. Now we are going to look at this process in reverse in order to factor expressions.

Factor with integers

There are many ways to find the greatest common factor between two numbers, such as looking at the prime factorizations or listing all factors. With practice, you will be able to identify the GCF more quickly.

To factor algebraic equations, we need to find the greatest common factors (GCF) between the terms, making sure to consider both the numeric values and algebraic terms.

Examples

Example 1

Factor 40x-25.

Worked Solution
Create a strategy

We will first find the GCF of 40x and -25. We can then rewrite each expression as a product of the GCF and any remaining factors and then factor out the GCF.

Apply the idea

The x factor is not common between 40x and -25, so we are looking for a numeric common factor. We need to consider factors of 40 and -25 to find the greatest common factor between them.

Using the listing method, the factors of 40 and 25 are:

  • 40: 1,\,2,\,4,\,5,\,8,\,10,\,20,\,40

  • 25: 1,\,5,\,25

Using prime factorization:

  • 40=2\times 2 \times 2 \times 5

  • 25=5\times 5

Both the listing method and prime factorization show that the greatest common factor is 5.

With the GCF of 5, we need to break 40x and -25 into a product with 5.

  • 40x=5(8x)

  • -25=5(-5)

40x-25=5(8x-5)

Reflect and check

We can always check to see if we have factored correctly by distributing it back out again.

\displaystyle 5(8x-5)\displaystyle =\displaystyle 5\times 8x+5\times(-5)
\displaystyle =\displaystyle 40x+(-25)
\displaystyle =\displaystyle 5\times 8x+5\times(-5)

Example 2

Factor: 4xyz - 24wxz

Worked Solution
Create a strategy

Find the greatest common numerical factor then find the common algebraic factor.

Apply the idea

The greatest common numerical factor is 4. The common algebraic factor is xz. So the GCF is 4xz.

  • 4xyz=4xz(y)

  • -24wxz=4xz(-6w)

\displaystyle 4xyz-24wxz\displaystyle =\displaystyle 4xz(y)+4xz(-6w)
\displaystyle =\displaystyle 4xz\left(y-6w\right)
Idea summary

Follow these steps for factoring out a GCF:

  1. Identify the GCF.

    This can be the greatest numerical factor or the common algebraic factor or both.
  2. Rewrite each term as a product of the GCF and the remaining factors.
  3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses.

Outcomes

7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

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