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Course: Algebra (all content) > Unit 10
Lesson 11: Factoring monomials- Which monomial factorization is correct?
- Factoring monomials
- Worked example: finding the missing monomial factor
- Worked example: finding missing monomial side in area model
- Factor monomials
- Greatest common factor of monomials
- Greatest common factor of monomials
- Greatest common factor of monomials
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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 , like . A polynomial is a sum of monomials, like .
If , then and are factors of , and is divisible by and . 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 .
Notice that when you multiply each expression on the right, you get .
Reflection question
Completely factoring monomials
Review: integer factorization
To factor an integer completely, we write it as a product of primes.
For example, we know that .
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 , we can write the prime factorization of as and write as . Therefore, this is the complete factorization of :
Check your understanding
Finding missing factors of monomials
Review: integer factorization
Suppose we know that for some integer . How can we find the other factor?
Well, we can solve the equation for by dividing both sides of the equation by . The missing factor is .
And now to monomials...
We can extend these ideas to monomials. For example, suppose for some monomial . We can find by dividing by :
We can check our work by showing that the product of and is indeed .
Check your understanding
A note about multiple factorizations
Consider the number . We can write four different factorizations of this number.
However, there is only one prime factorization of the number , i.e. .
The same idea holds with monomials. We can factor in many ways. Here are a few different factorizations.
Yet there is only one complete factorization!
Challenge problems
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