How to Factor by Grouping

PDF download Download Article
Learn to untangle quadratic equations so you can solve them correctly
PDF download Download Article

Grouping is a specific technique used to factor polynomial equations. You can use it with quadratic equations and polynomials that have four terms. The two methods are similar, but do vary slightly.

Method 1
Method 1 of 2:

Quadratic Equations

PDF download Download Article
  1. If you plan to use this method, the equation should follow a basic format of: ax 2 + bx + c. [1]
    • This process is usually used when the leading coefficient (the a term) is a number other than "1," but it can also be used for quadratic equations in which a = 1 .
    • Example: 2x 2 + 9x + 10
  2. Multiply the a term and c term together. The product of these two terms is referred to as the master product . [2]
    • Example: 2x 2 + 9x + 10
      • a = 2; c = 10
      • a * c = 2 * 10 = 20
    Advertisement
  3. List the factors of your master product, separating them into their natural pairs (the pairs required to produced the master product). [3]
    • Example: The factors of 20 are: 1, 2, 4, 5, 10, 20
      • Written in factor pairs: (1, 20), (2, 10), (4, 5)
  4. Look through the factor pairs and determine which set will produce the b term—the middle term and the coefficient of x —when added together. [4]
    • If your master product was negative, you will need to find a pair of factors that equal the b term when subtracted from one another.
    • Example: 2x 2 + 9x + 10
      • b = 9
      • 1 + 20 = 21; this is not the correct pair
      • 2 + 10 = 12; this is not the correct pair
      • 4 + 5 = 9; this is the correct pair
  5. Rewrite the center term, breaking it apart into the factor pair previously identified. Make sure that you include the proper signs (plus or minus). [5]
    • Note that the order of the center terms should not matter for this problem. No matter which order you write the terms in, the end result should be the same.
    • Example: 2x 2 + 9x + 10 = 2x 2 + 5x + 4x + 10
  6. Group the first two terms into a pair and the second two terms into a pair. [6]
    • Example: 2x 2 + 5x + 4x + 10 = (2x 2 + 5x) + (4x + 10)
  7. Find the common factors of the pair and factor them out. Rewrite the equation accordingly. [7]
    • Example: x(2x + 5) + 2(2x + 5)
  8. There should be a shared binomial parentheses between the two halves. Factor this out, and place the other terms in another parentheses. [8]
    • Example: (2x + 5)(x + 2)
  9. You should now have your final answer. [9]
    • Example: 2x 2 + 9x + 10 = (2x + 5)(x + 2)
      • The final answer is: (2x + 5)(x + 2)
    10. Advertisement

Additional Examples

  1. 4x 2 - 3x - 10
    • a * c = 4 * -10 = -40
    • Factors of 40: (1, 40), (2, 20), (4, 10), (5, 8)
    • Correct factor pair: (5, 8); 5 - 8 = -3
    • 4x 2 - 8x + 5x - 10
    • (4x 2 - 8x) + (5x - 10)
    • 4x(x - 2) + 5(x - 2)
    • (x - 2)(4x + 5)
  2. 8x 2 + 2x - 3
    • a * c = 8 * -3 = -24
    • Factors of 24: (1, 24), (2, 12), (4, 6)
    • Correct factor pair: (4, 6); 6 - 4 = 2
    • 8x 2 + 6x - 4x - 3
    • (8x 2 + 6x) - (4x + 3)
    • 2x(4x + 3) - 1(4x + 3)
    • (4x + 3)(2x - 1)
    3. Advertisement
Method 2
Method 2 of 2:

Polynomials with Four Terms

PDF download Download Article
  1. The equation should have four separate terms. The exact appearance of those four terms can vary, however. [10]
    • Usually, you will use this method when you see a polynomial equation that looks like: ax 3 + bx 2 + cx + d
    • The equation may also look like:
      • axy + by + cx + d
      • ax 2 + bx + cxy + dy
      • ax 4 + bx 3 + cx 2 + dx
      • Or similar variations.
    • Example: 4x 4 + 12x 3 + 6x 2 + 18x
  2. Determine if all four terms have anything in common. The greatest common factor among the four terms, if any common factors exist, should be factored out of the equation. [11]
    • If the only thing all four terms has in common is the number "1," there is no GCF and nothing can be factored out at this point.
    • When you do factor out a GCF, make sure that you continue to keep it at the front of your equation as you work. This factored out GCF must be included as part of your final answer for that answer to be accurate.
    • Example: 4x 4 + 12x 3 + 6x 2 + 18x
      • Each term has 2x in common, so the problem can be rewritten as:
      • 2x(2x 3 + 6x 2 + 3x + 9)
  3. Group the first two terms together and the second two terms together. [12]
    • If the first term of the second group has a minus sign in front of it, you will need to put a minus sign in front of the second parentheses. You will need to change the sign of the second term in that grouping to reflect that choice.
    • Example: 2x(2x 3 + 6x 2 + 3x + 9) = 2x[(2x 3 + 6x 2 ) + (3x + 9)]
  4. Identify the GCF in each binomial pair and factor it to the outside of the pair. Rewrite the equation accordingly. [13]
    • At this point, you might be faced with a choice between factoring out a positive number or a negative number for the second group. Look at the signs before the second and fourth terms.
      • When the two signs are the same (both positive or both negative), factor out a positive number.
      • When the two signs are different (one negative and one positive), factor out a negative number.
    • Example: 2x[(2x 3 + 6x 2 ) + (3x + 9)] = 2x 2 [2x 2 (x + 3) + 3(x + 3)]
  5. The binomial pair inside both parentheses should be the same. Factor this out of the equation, then group the remaining terms into another parentheses set. [14]
    • If the binomials inside the current sets of parentheses do not match, double-check your work or try rearranging your terms and grouping the equation again.
    • The parentheses must match. If they do not match no matter what you try, the problem cannot be factored by grouping or by any other method.
    • Example: 2x 2 [2x 2 (x + 3) + 3(x + 3)] = 2x 2 [(x + 3)(2x 2 + 3)]
  6. You should have the final answer at this point.
    • Example: 4x 4 + 12x 3 + 6x 2 + 18x = 2x 2 (x + 3)(2x 2 + 3)
      • The final answer is: 2x 2 (x + 3)(2x 2 + 3)
    7. Advertisement

Additional Examples

  1. 6x 2 + 2xy - 24x - 8y
    • 2[3x 2 + xy - 12x - 4y]
    • 2[(3x 2 + xy) - (12x + 4y)]
    • 2[x(3x + y) - 4(3x + y)]
    • 2[(3x + y)(x - 4)]
    • 2(3x + y)(x – 4)
  2. x 3 - 2x 2 + 5x - 10
    • (x 3 - 2x 2 ) + (5x - 10)
    • x 2 (x - 2) + 5(x - 2)
    • (x - 2)(x 2 + 5)
  3. 3
    When factoring by grouping, after factoring out the GCF from each group, the binomials in the two sets of parentheses must match up in order to go any further with the problem. [15]
    4. Advertisement

Community Q&A

Search
Add New Question
  • Question
    What is the factor of 3xy - zw + 3xw?
    Donagan
    Top Answerer
    As a first-degree polynomial, that expression cannot be factored, because there is no letter or number common to all three terms.
    Thanks! We're glad this was helpful.
    Thank you for your feedback.
    If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
    Yes No
    Not Helpful 6 Helpful 8
  • Question
    How do you factor if the coefficient b is greater than (a)(c)? Example, 2x² - 10x +4.
    Community Answer
    In this example, you can factor out a 2, but it cannot be factored further.
    Thanks! We're glad this was helpful.
    Thank you for your feedback.
    If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
    Yes No
    Not Helpful 0 Helpful 2
  • Question
    How do you do ax+ay+bx + by −cx−cy using the grouping method?
    Community Answer
    That equals a(x + y) + b(x + y) - c(x + y) = (a + b - c)(x + y). By the way, here's how you say that final expression: "the quantity a plus b minus c, times the quantity x plus y."
    Thanks! We're glad this was helpful.
    Thank you for your feedback.
    If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
    Yes No
    Not Helpful 0 Helpful 0
Ask a Question
200 characters left
Include your email address to get a message when this question is answered.
Submit
      Advertisement

      Video

      Tips

      Submit a Tip
      All tip submissions are carefully reviewed before being published
      Name
      Please provide your name and last initial
      Submit
      Thanks for submitting a tip for review!

      You Might Also Like

      How to
      Find the Maximum or Minimum Value of a Quadratic Function Easily
      How to
      Solve a Cubic Equation
      How to
      Complete the Square to Solve a Quadratic Equation
      How to
      Factor a Cubic Polynomial
      How to
      Learn Algebra
      How to
      Solve for X
      Armor Class in D&D 5E: What Is It & How Can You Calculate It?
      Your Guide to Flipping the Inequality Sign (And Solving Inequalities)
      How to
      Algebraically Find the Intersection of Two Lines
      How to
      Solve Systems of Algebraic Equations Containing Two Variables
      How to
      Solve a Quadratic Equation: A Step-by-Step Guide
      How to
      Calculate Frequency
      How to
      Find the Inverse of a Function
      How to
      Solve Systems of Equations
      Advertisement

      References

      1. ↑ https://www.mathwarehouse.com/algebra/polynomial/factor-trinomial-when-a-is-not-1.php
      2. ↑ https://www.mathwarehouse.com/algebra/polynomial/factor-trinomial-when-a-is-not-1.php
      3. ↑ https://mathbitsnotebook.com/Algebra1/Factoring/FCGrouping.html
      4. ↑ https://www.mathsisfun.com/algebra/factoring-quadratics.html
      5. ↑ https://www.purplemath.com/modules/simpfact3.htm
      6. ↑ https://www.purplemath.com/modules/simpfact3.htm
      7. ↑ https://www.mathsisfun.com/algebra/factoring-quadratics.html
      8. ↑ https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-quadratics-grouping/a/factoring-by-grouping
      9. ↑ https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-quadratics-grouping/a/factoring-by-grouping
      1. ↑ https://courses.lumenlearning.com/wm-developmentalemporium/chapter/5-2-1-factor-trinomials/
      2. ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut27_gcf.htm
      3. ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut27_gcf.htm
      4. ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm
      5. ↑ https://courses.lumenlearning.com/wm-developmentalemporium/chapter/5-2-1-factor-trinomials/
      6. ↑ https://math.libretexts.org/Courses/Monroe_Community_College/MTH_104_Intermediate_Algebra/6%3A_Factoring/6.2%3A_Greatest_Common_Factor_and_Factor_by_Grouping

      About This Article

      Article Summary X

      To factor a quadratic equation by grouping, start by multiplying the "a" term by the "c" term to get the master product. Then, list all of the factors of your master product, and separate them into their natural pairs. Next, look for the factor pair that has a sum equal to the "b" term in the equation, and split the "b" term into 2 factors. Finally, group the terms to form pairs, factor out each pair, and factor out the shared parentheses. To learn how to factor polynomials by grouping, scroll down!

      Did this summary help you?
      In other languages
      Thanks to all authors for creating a page that has been read 191,676 times.

      Reader Success Stories

      • Jacob B.

        Jul 12, 2020

        "I've done a couple of these before, but I had since forgotten how, and this definitely helped jog my memory. ..." more
      Share your story

      Did this article help you?

      Advertisement