Factoring Expressions With Exponents

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  • 0:03 Factoring Expressions…
  • 0:42 Example 1
  • 2:30 Example 2
  • 4:03 Example 3
  • 5:31 Lesson Summary
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Lesson Transcript
Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson offers a detailed tutorial on how to factor polynomial expressions with exponents. The greatest common factor is used to factor these expressions.

Factoring Expressions with Exponents

Think of factoring an expression with exponents as dividing that expression by one of its factors. A factor of an expression is a number or expression that divides into the expression evenly. For example, the number 24 has many factors; 1, 2, 3, 4, 6, 8, 12, and 24 are all factors of 24 because they can divide 24 evenly with no remainder.

The greatest common factor, or GCF, of a set of numbers or an expression is the largest number or expression that divides evenly into all of the numbers or terms of the expression.

Example 1

Let's look at an example to see how this works. We'll take the expression:

example 1

To factor, we first must look for the greatest common factor of each term in the expression. There are two things we're going to look at: the coefficient and the exponents of the variables. A coefficient is the number that is multiplied times the variable, or the number in front of the variable. Variables are the letters in the expression. And an exponent is the power of the variable.

In this example, the coefficients are 3 and 12, which have a greatest common factor of 3 since it's the largest number that divides into both numbers evenly.

Second, we look at the variables and, more specifically, the exponents of the variables. In this example, the exponents are 3 and 2. To find the greatest common factor of the variables, we take the lowest exponent, which is 2.

Thus, the greatest common factor of these two terms is:


To factor, we must divide the original expression by the greatest common factor:


To divide, we follow two steps:

  • First, we divide the numbers: When we divide 3 by 3, we get 1. When we divide 12 by 3, we get 4.

  • Second, we subtract the exponents: When we subtract 2 from 3, we get 1. When we subtract 2 from 2, we get zero. That zero means that there is no longer a variable left. After doing these two steps, we have x minus 4.

Then, to factor, we first write the GCF. In parentheses, we put the quotient that we found when we divided the original expression by the GCF. This means we have:


Example 2

Let's work through another example. We'll look at:

example 2

  • First, we find the greatest common factor of the two coefficients. The GCF of 4 and 6 is 2.

  • Next, we look at the exponents. The lowest exponent of x is 5, and the lowest exponent of y is 4. So, the GCF of our expression is:


Now, to factor, we divide the original expression by the GCF:


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