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Factoring monomials

Learn how to completely factor monomial expressions, or find the missing factor in a monomial factorization.

What you should be familiar with before this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of x, like 3x2. A polynomial is a sum of monomials, like 3x2+6x1.
If A=BC, then B and C are factors of A, and A is divisible by B and C. To review this material, check out our article on Factoring and divisibility.

What you will learn in this lesson

In this lesson, you will learn how to factor monomials. You will use what you already know about factoring integers to help you in this quest.

Introduction: What is monomial factorization?

To factor a monomial means to express it as a product of two or more monomials.
For example, below are several possible factorizations of 8x5.
  • 8x5=(2x2)(4x3)
  • 8x5=(8x)(x4)
  • 8x5=(2x)(2x)(2x)(x2)
Notice that when you multiply each expression on the right, you get 8x5.

Reflection question

Andrei, Amit and Andrew were each asked to factor the term 20x6 as the product of two monomials. Their responses are shown below.
1) Which of the students factored 20x6 correctly?
Choose all answers that apply:

Completely factoring monomials

Review: integer factorization

To factor an integer completely, we write it as a product of primes.
For example, we know that 30=235.

And now to monomials...

To factor a monomial completely, we write the coefficient as a product of primes and expand the variable part.
For example, to completely factor 10x3, we can write the prime factorization of 10 as 25 and write x3 as xxx. Therefore, this is the complete factorization of 10x3:

Check your understanding

2) Which of the following is the complete factorization of 6x2?
Choose 1 answer:

3) Which of the following is the complete factorization of 14x4?
Choose 1 answer:

Finding missing factors of monomials

Review: integer factorization

Suppose we know that 56=8b for some integer b. How can we find the other factor?
Well, we can solve the equation 56=8b for b by dividing both sides of the equation by 8. The missing factor is 7.

And now to monomials...

We can extend these ideas to monomials. For example, suppose 8x5=(4x3)(C) for some monomial C. We can find C by dividing 8x5 by 4x3:
8x5=(4x3)(C)8x54x3=(4x3)(C)4x3Divide both sides by 4x32x2=CSimplify with properties of exponents
We can check our work by showing that the product of 4x3 and 2x2 is indeed 8x5.

Check your understanding

4) Find the missing factor B that makes the following equality true.
Choose 1 answer:

5) Find the missing factor C that makes the following equality true.

A note about multiple factorizations

Consider the number 12. We can write four different factorizations of this number.
  • 12=26
  • 12=34
  • 12=121
  • 12=223
However, there is only one prime factorization of the number 12, i.e. 223.
The same idea holds with monomials. We can factor 18x3 in many ways. Here are a few different factorizations.
  • 18x3=29x3
  • 18x3=36xx2
  • 18x3=233x3
Yet there is only one complete factorization!

Challenge problems

6*) Write the complete factorization of 22xy2.

7*) The rectangle below has an area of 24x3 square meters and a length of 4x2 meters.
What is the width of the rectangle?

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