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Factoring by common factor review

The expression 6m+15 can be factored into 3(2m+5) using the distributive property. More complex expressions like 44k^5-66k^4 can be factored in much the same way. This article provides a couple of examples and gives you a chance to try it yourself.

Example 1

Factor.
6, m, plus, 15
Both terms share a common factor of start color #e07d10, 3, end color #e07d10, so we factor out the start color #e07d10, 3, end color #e07d10 using the distributive property:
6m+15=3(2m+5)\begin{aligned} &6m+15\\\\ =&\goldD{3}(2m+5) \end{aligned}
Want a more in-depth explanation? Check out this video.

Example 2

Factor out the greatest common monomial.
44, k, start superscript, 5, end superscript, minus, 66, k, start superscript, 4, end superscript, plus, 77, k, cubed
The coefficients are 44, comma, 66, comma and 77, and their greatest common factor is start color #11accd, 11, end color #11accd.
The variables are k, start superscript, 5, end superscript, comma, k, start superscript, 4, end superscript, comma and k, cubed, and their greatest common factor is start color #11accd, k, cubed, end color #11accd.
Therefore, the greatest common monomial factor is start color #11accd, 11, k, cubed, end color #11accd.
Factoring, we get:
44k566k4+77k3=11k3(4k2)+11k3(6k)+11k3(7)=11k3(4k26k+7)\begin{aligned} &44k^5-66k^4+77k^3\\\\ =&\blueD{11k^3}(4k^2)+\blueD{11k^3}(-6k)+\blueD{11k^3}(7)\\\\ =&\blueD{11k^3}(4k^2-6k+7) \end{aligned}
Want another example like this one? Check out this video.

Practice

Factor the polynomial below by its greatest common monomial factor.
3, b, start superscript, 5, end superscript, plus, 15, b, start superscript, 4, end superscript, minus, 18, b, start superscript, 7, end superscript, equals

Want more practice? Check out this exercise.

Want to join the conversation?

  • spunky sam blue style avatar for user 💎Chυcκ Lørrε💎
    I heard there's a way that can solve all the polynomials speedy. That's cross-factoring, but anyone knows how to use cross-factoring, and could it really solve all kinds of polynomials speedy?
    (3 votes)
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    • mr pink red style avatar for user David Lee
      I guess the term 'cross-factoring' is used when you're dividing a polynomial by a polynomial. There is a term 'cross out' when simplifying a polynomial. You just need to factor the denominator and numerator. Then, find the same factors and divide both numerator and denominator. We usually call this 'cross out'.
      Hope this help! If you have any questions or need help, please ask! :)
      (3 votes)
  • area 52 purple style avatar for user diamond  krigger
    my test is today and im still struggling
    (2 votes)
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  • mr pants purple style avatar for user Bhavya
    I still am having a little bit of trouble but I think your video helped. If you can explain a little more I would rely think that it would be helpful?
    (1 vote)
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  • primosaur seedling style avatar for user Makenna Petrusek
    I think you don't have to rewrite the whole equation with the answer
    (1 vote)
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  • starky tree style avatar for user emiliamolla10
    a^2b-ab^2
    can you help me solve this?
    (1 vote)
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  • blobby green style avatar for user Esteban Villegas
    How do you facto when all the numbers are negative?
    (1 vote)
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  • leaf yellow style avatar for user pearl.herrera
    i still dont know or understand how to do this
    (1 vote)
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  • blobby green style avatar for user acepuglisi5
    what's your favorite color? because I really need to know
    (1 vote)
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  • ohnoes default style avatar for user wayne
    I don't get what it means by "Factor the polynomial as the product of two binomials". Does it mean to factor it out after the two are multiplied or does it mean to have the product be the factor?
    (1 vote)
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    • aqualine ultimate style avatar for user AD Baker
      Wayne,
      Factor the polynomial as the product of two binomials mean that you are asked to take an expression that looks like this

      a^2+2ab+b^2 (a polynomial)

      and algebraically manipulate the terms until the expression looks like this:

      (a+b)(a+b) two binomial factors being multiplied
      (1 vote)
  • aqualine seed style avatar for user Carevee
    The third term's sign is a negative, right? How is the trinomial's Greatest Common Factor a positive, when one term is a negative?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Factoring is correct as shown on the screen. Factoring out a common factor divides each term by that factor to remove the common factor from each term. Thus, the signs of the common factor and the new terms follow the rules for division with signed numbers.
      positive / positive = positive
      positive / negative = negative
      negative / positive = negative
      negative / negative = positive.

      The common factors "11k^3" is positive. The third term "77k^3" is also positive.
      A positive / a positive = a positive.
      Hope this helps.
      (1 vote)