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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.
AndreiAmitAndrew
20x6=(2x)(10x5)20x6=(4x3)(5x3)20x6=(20x2)(x3)
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:
10x3=25xxx

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.
(4x3)(2x2)=42x3x2=8x5

Check your understanding

4) Find the missing factor B that makes the following equality true.
28x5=(B)(7x)
Choose 1 answer:

5) Find the missing factor C that makes the following equality true.
40x9=(C)(4x3)
C=

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!
18x3=233xxx

Challenge problems

6*) Write the complete factorization of 22xy2.
22xy2=

7*) The rectangle below has an area of 24x3 square meters and a length of 4x2 meters.
A rectangle with the width labeled width and the length being four x squared. Inside the rectangle is twenty four x cubed.
What is the width of the rectangle?
Width=
meters

Want to join the conversation?

  • starky sapling style avatar for user Jannete
    i need help with this i have already tried a lot but i just don't get it.
    (18 votes)
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    • duskpin ultimate style avatar for user LarissaAnne
      Let me see if I can help.

      One way to look at factorizing monomials is pulling them apart into their basic parts. Like you had a peanut butter and jelly sandwich. Pulling that into its basic parts, you could get bread times bread times peanut butter times jelly. That would be its complete factorization.

      Now for a monomial

      Say you have a monomial (or one term) that is 24x^2. There are a few ways to factor this. One example would be: (4x)(6x) This is not a complete factorization, but it is still a valid way to factorize it. Another way to do it would be (3)(8x^2) Its complete factorization would be 2 times 2 times 2 times 3 times x times x.

      When factoring, you just need to make sure that when you preform all the operations, they still come out to be the original number. Like I need to make sure (4x)(6x) = 24x^2. Let's check. 4 times 6 equals 24, and x times x equals x^2. Those two multiplied together equals 24x^2.

      I hope that helped! :)
      (41 votes)
  • leafers tree style avatar for user Neon Froggie
    You don't earn energy points on the articles, but you guys do it anyway. Good job! Keep it up.
    (16 votes)
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  • blobby blue style avatar for user Heavo
    I'm so glad that he went into more detail with more videos. Because in the first video I was so lost lol.
    (14 votes)
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  • duskpin tree style avatar for user Cassandra Lehnerd
    I'm having a hard time understanding how to find the width of the rectangle,how should i go about finding the width of a rectangle?
    (7 votes)
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  • starky seed style avatar for user Katelyn Friis
    I am still confused on how to tell if it is a monomial or not. Can you help me?
    (8 votes)
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  • stelly blue style avatar for user obiudeozo0
    If it has 1 term, it is a monomial. If it has 2 terms, it is a binomial. 3 terms or more is a polynomial.
    (4 votes)
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  • blobby green style avatar for user Stephanie Gonzalez
    How can understand this better?
    (1 vote)
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  • blobby green style avatar for user drayush.sr
    this helped a lot but is there anyway to understand it better
    (4 votes)
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  • blobby green style avatar for user xgibbks
    idk what most of these are
    (4 votes)
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    • blobby green style avatar for user paradise
      Well, one way you can complete factorization is when you list the prime factors (they can't be simplified anymore) along with the amount of x's. For example, the complete factorization of 64x^6 would work like this:
      64 = 8*8 = 2*2*2*2*2*2 (6 2's)
      so you would write the complete factorization as: 2*2*2*2*2*2*x*x*x*x*x*x, this is 6 2's and 6 x's.
      A complete factorization is one where the factors cannot be simplified anymore, like shown above. In addition, there is only one complete factorization for a given monomial.

      If you don't have to completely factor the monomial, you can first find factors of the coefficient and then split the x's. For example, if I had 64x^6 again, I could list out the factors of 64 (coefficient) first.
      64: 1 & 64, 2 & 32, 4 & 16, 8 & 8
      You could then pick any of these pairs. Let's say we use 4 and 16. Then, we factor x^6, we could turn this into any two pairs that add up to 6. This is because of the exponent multiplication method. When you have the same base (x), you can add the exponents.
      6 = 1+5, 2+4, 3+3. Let's say we choose 3+3.
      Now, we can write it out. Recall that our coefficients are 4 and 16.
      We can write it out like this: 4x^3 * 16x^3.
      If we check it, we can multiply the coefficients (16*4 = 64) and the x's (x^3 * x^3 = x^6) and we get 64x^6, so this factorization is correct. Note that this is not the only factorization of 64x^6, you could select any other group of x's and any other coefficients that work.

      I hope this helps and let me know if you have any questions! :)
      P.S. sorry for the late reply
      (1 vote)
  • blobby green style avatar for user coco1max2
    philosophically why do we simplify?
    (2 votes)
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    • blobby green style avatar for user matthew.weltmer
      if someone told you, "I want 2 bottles of water, then 5 bottles of water, then one bottle of water, then 1 bottle of water!" and you say, "whats that total?" your asking them to simplify because its easier.

      sim·pli·fy
      /ˈsimpləˌfī/
      verb
      make (something) simpler or easier to do or understand.
      "an overhaul of court procedure to simplify litigation"
      (1 vote)