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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 3, x, squared. A polynomial is a sum of monomials, like 3, x, squared, plus, 6, x, minus, 1.
If A, equals, B, dot, C, 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 8, x, start superscript, 5, end superscript.
  • 8, x, start superscript, 5, end superscript, equals, left parenthesis, 2, x, squared, right parenthesis, left parenthesis, 4, x, cubed, right parenthesis
  • 8, x, start superscript, 5, end superscript, equals, left parenthesis, 8, x, right parenthesis, left parenthesis, x, start superscript, 4, end superscript, right parenthesis
  • 8, x, start superscript, 5, end superscript, equals, left parenthesis, 2, x, right parenthesis, left parenthesis, 2, x, right parenthesis, left parenthesis, 2, x, right parenthesis, left parenthesis, x, squared, right parenthesis
Notice that when you multiply each expression on the right, you get 8, x, start superscript, 5, end superscript.

Reflection question

Andrei, Amit and Andrew were each asked to factor the term 20, x, start superscript, 6, end superscript as the product of two monomials. Their responses are shown below.
AndreiAmitAndrew
20, x, start superscript, 6, end superscript, equals, left parenthesis, 2, x, right parenthesis, left parenthesis, 10, x, start superscript, 5, end superscript, right parenthesis20, x, start superscript, 6, end superscript, equals, left parenthesis, 4, x, cubed, right parenthesis, left parenthesis, 5, x, cubed, right parenthesis20, x, start superscript, 6, end superscript, equals, left parenthesis, 20, x, squared, right parenthesis, left parenthesis, x, cubed, right parenthesis
1) Which of the students factored 20, x, start superscript, 6, end superscript correctly?
Choose all answers that apply:
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, equals, 2, dot, 3, dot, 5.

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 10, x, cubed, we can write the prime factorization of 10 as 2, dot, 5 and write x, cubed as x, dot, x, dot, x. Therefore, this is the complete factorization of 10, x, cubed:
10, x, cubed, equals, 2, dot, 5, dot, x, dot, x, dot, x

Check your understanding

2) Which of the following is the complete factorization of 6, x, squared?
Choose 1 answer:
Choose 1 answer:

3) Which of the following is the complete factorization of 14, x, start superscript, 4, end superscript?
Choose 1 answer:
Choose 1 answer:

Finding missing factors of monomials

Review: integer factorization

Suppose we know that 56, equals, 8, b for some integer b. How can we find the other factor?
Well, we can solve the equation 56, equals, 8, b 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 8, x, start superscript, 5, end superscript, equals, left parenthesis, 4, x, cubed, right parenthesis, left parenthesis, C, right parenthesis for some monomial C. We can find C by dividing 8, x, start superscript, 5, end superscript by 4, x, cubed:
8x5=(4x3)(C)8x54x3=(4x3)(C)4x3Divide both sides by 4x32x2=CSimplify with properties of exponents\begin{aligned}8x^5&=(4x^3)(C)\\ \\ \dfrac{8x^5}{4x^3}&=\dfrac{(4x^3)(C)}{4x^3}&&\small{\gray{\text{Divide both sides by }4x^3}}\\ \\\\\\ 2x^2&=C&&\small{\gray{\text{Simplify with properties of exponents}}} \end{aligned}
We can check our work by showing that the product of 4, x, cubed and 2, x, squared is indeed 8, x, start superscript, 5, end superscript.
(4x3)(2x2)=42x3x2=8x5\begin{aligned}(\purpleC{4}\tealD {x^3})(\purpleC{2}\tealD{x^2})&=\purpleC 4\cdot \purpleC{2}\cdot \tealD {x^3}\cdot \tealD{x^2}\\ \\ &=\purpleC{8}\tealD{x^5} \end{aligned}

Check your understanding

4) Find the missing factor B that makes the following equality true.
28, x, start superscript, 5, end superscript, equals, left parenthesis, B, right parenthesis, left parenthesis, 7, x, right parenthesis
Choose 1 answer:
Choose 1 answer:

5) Find the missing factor C that makes the following equality true.
40, x, start superscript, 9, end superscript, equals, left parenthesis, C, right parenthesis, left parenthesis, 4, x, cubed, right parenthesis
C, equals

A note about multiple factorizations

Consider the number 12. We can write four different factorizations of this number.
  • 12, equals, 2, dot, 6
  • 12, equals, 3, dot, 4
  • 12, equals, 12, dot, 1
  • 12, equals, 2, dot, 2, dot, 3
However, there is only one prime factorization of the number 12, i.e. 2, dot, 2, dot, 3.
The same idea holds with monomials. We can factor 18, x, cubed in many ways. Here are a few different factorizations.
  • 18, x, cubed, equals, 2, dot, 9, dot, x, cubed
  • 18, x, cubed, equals, 3, dot, 6, dot, x, dot, x, squared
  • 18, x, cubed, equals, 2, dot, 3, dot, 3, dot, x, cubed
Yet there is only one complete factorization!
18, x, cubed, equals, 2, dot, 3, dot, 3, dot, x, dot, x, dot, x

Challenge problems

6*) Write the complete factorization of 22, x, y, squared.
22, x, y, squared, equals

7*) The rectangle below has an area of 24, x, cubed square meters and a length of 4, x, squared meters.
What is the width of the rectangle?
start text, W, i, d, t, h, end text, equals
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.
    (10 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! :)
      (22 votes)
  • 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?
    (5 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?
    (5 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.
    (2 votes)
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  • winston baby style avatar for user Italian Pizza
    i hate khan acadeemy no ofence but i dont learn a thing
    (0 votes)
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  • blobby green style avatar for user coco1max2
    philosophically why do we simplify?
    (1 vote)
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    • mr pants purple style avatar for user The Purple Bear
      I don't know about the philosophy but I do know why we do simplification. Basically, it makes the math waayyy less hairy. I believe that 932/1864 * 3/12 is harder to solve than 1/2 * 1/4. :o. it also makes sure everyone has the same answer! If you don't simplify, you might get totally different numbers that don't look at all even close, 345/690 and 17/34 don't look related. however, if you simplified, you would know they are indeed the same number, 1/2
      (3 votes)
  • duskpin ultimate style avatar for user Joseph Gonzalez
    So a simple way of doing it is by dividing the total by the number given and then subtracting the exponent given from the total and then what you get is your answer?
    (2 votes)
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    • orange juice squid orange style avatar for user Jonathan
      Technically, yes. A better way to think about it is to think that you are dividing the coefficients and the exponents separately.

      For example, let's say we have to find A in (A)4x^2=8x^3. We would divide both sides by 4x^2. First, divide the coefficients: 8/4=2. Then, divide the exponents: x^3/x^2=x*x*x/x*x=x. Multiplying our two results together will give us our answer: 2x.

      Hope this helps!

      Jonathan Myung
      (0 votes)
  • aqualine ultimate style avatar for user ColinTCline
    Isn't 3x^2+6x-1 a trinomial instead of a polynomial?
    (0 votes)
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  • blobby green style avatar for user sai.jeenagala
    A monomial is a multiplication with 1 number.
    (1 vote)
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  • aqualine seed style avatar for user MenthuPlayz
    i don't get the last two
    (0 votes)
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    • mr pink green style avatar for user David Severin
      6) just asks for the prime factorization, so 22 = 2 • 11, so just add that to the variables to get 2 • 11 x • y • y. Prime factorization helps us find GCF and LCM
      7) this is popular way to create problems, note that A = l w, so we can set an equation
      24 x^3 = 4 x^2 w divide both sides by 4 x^2, we get 4 • 6 xxx/4 xx and canceling we get 6x.
      (3 votes)