Math resources Algebra Factoring

Factor by grouping

Factor by grouping

Here you will learn how to factor polynomials with a strategy called factor by grouping.

Students first learn how to factor by grouping in high school.

What is factor by grouping?

Factor by grouping is writing the polynomial as a product of its factors. It is the inverse process of multiplying algebraic expressions using the distributive property.

There are several strategies for factoring polynomials. This page will overview the strategy factor by grouping for polynomial equations.

For example,

Factor this four-term polynomial by grouping:

x^2+x+3x+3

Group the first two terms together and the second two terms together.

Factor by grouping 1 US

Factor out the greatest common factor (GCF) in the first group and the second group.

Factor by grouping 2 US

x(x+1)+3(x+1)

Notice how both sets of parentheses are the same. When factoring by grouping, you want the binomials to be the same after factoring out the GCF.

Looking at the expression, the common binomial is (x+1) so like the GCF, it can be factored out.

x^2+x+3 x+3=(x+1)(x+3)

What is factor by grouping?

What is factor by grouping?

Common Core State Standards

How does this relate to high school math?

  • Algebra – Seeing Structure in Equations (HSA-SSE.B.3a)
    Factor a quadratic expression to reveal the zeros of the function it defines.

[FREE] Factoring By Grouping Worksheet (High School)

[FREE] Factoring By Grouping Worksheet (High School)

[FREE] Factoring By Grouping Worksheet (High School)

Use this worksheet to check your high school students’ understanding of factoring by grouping. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE
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[FREE] Factoring By Grouping Worksheet (High School)

[FREE] Factoring By Grouping Worksheet (High School)

[FREE] Factoring By Grouping Worksheet (High School)

Use this worksheet to check your high school students’ understanding of factoring by grouping. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

How to factor algebraic expressions by grouping

In order to factor by grouping:

  1. Group the first two terms together and the last two terms together.
  2. Factor out the \textbf{GCF} from each binomial.
  3. Factor out the common binomial.

Factor by grouping examples

Example 1: factoring a quadratic equation

Factor the expression: x^2-8 x-2 x+16

  1. Group the first two terms together and the last two terms together. 

Group the terms that have common factors. In this case, it is the first term with the second term and the third term with the fourth term. Use parentheses to show the groupings.

\left(x^2-8x\right)+(-2x+16)

2Factor out the \textbf{GCF} from each binomial.

In x^2-8x , both terms have a factor of x, so x^2-8x=x(x-8).

In -2x+16, both terms have a factor of 2, so -2x+16=2(-x+8).

Note here that factoring out the positive coefficient creates the binomial -x+8 but as the binomial from the previous two terms is x-8 so factor the negative coefficient to match the binomials.

Factor -2 from -2x+16, so -2x+16=-2(x-8).

3Factor out the common binomial.

\begin{aligned}&x^2-8x-2x+16\\\\&=\left(x^2-8x\right)+(-2x+16)\\\\&=x(x-8)+-2(x-8)\\\\&=(x-8)(x-2)\end{aligned}

Notice what happens when you use the commutative property before factoring a quadratic equation:

\begin{aligned}& x^2-8 x-2 x+16 \\\\& =x^2-2 x-8 x+16 \\\\& =x(x-2)-8(x-2) \\\\& =(x-8)(x-2)\end{aligned}

This shows us that we can group the x^2 term with either x term and the other x term with the constant.

Example 2: factoring a quadratic equation with a leading coefficient

Factor the expression: 3x^2+6 x+12 x+24

Group the first two terms together and the last two terms together.

Factor out the \textbf{GCF} from each binomial.

Factor out the common binomial.

Example 3: factoring a quadratic equation with a leading coefficient

Factor the expression: 2x^2-10 x+3 x-15

Group the first two terms together and the last two terms together.

Factor out the \textbf{GCF} from each binomial.

Factor out the common binomial.

Example 4: fraction coefficients

Factor the expression: x^2+\cfrac{1}{2} x+2x+1

Group the first two terms together and the last two terms together.

Factor out the \textbf{GCF} from each binomial.

Factor out the common binomial.

Example 5: factoring trinomials

Factor the expression: x^2-7 x-30

Group the first two terms together and the last two terms together.

Factor out the \textbf{GCF} from each binomial.

Factor out the common binomial.

Example 6: factoring a cubic function

Factor the expression: 4x^3-x^2+32x^2-8x

Group the first two terms together and the last two terms together.

Factor out the \textbf{GCF} from each binomial.

Factor out the common binomial.

Teaching tips for factoring

  • As an intro to this topic, review the distributive property with simple rational expressions and algebraic expressions. Since the factor by grouping method relies on GCF, it would be a good idea to review this topic as well.

  • Start by factoring lower degree polynomials, before advancing to higher degree polynomials.

  • Require students to check their work in some way. For example, they could use another strategy to factor or let x equal a value, such as x=5, and solve both the original expression and the factored expression. For the second strategy, encourage students to avoid using zeros or ones as the value for x, as this can sometimes lead to incorrect conclusions.

  • For struggling students, provide students with step-by-step guides or tutorials to refer back to when solving on their own.

Easy mistakes to make

  • Not finding common factors for both term groupings, particularly when the terms involve subtraction
    For example,
    x^2-8 x-2 x+16
    =\left(x^2-8 x\right)+(-2 x+16)
    =x(x-8)+2(-x+8) *Factoring out 2, does not create a common factor within the parentheses
    =x(x-8)-2(x-8) *Factoring out –2, does create a common factor
    =(x-8)(x-2)

  • Not using the most efficient strategy to factor a polynomial
    While it is important to expose students to the factor by grouping strategy, in the long run, students need to learn to decide which factoring strategy (such as factor by grouping, difference of squares or perfect square or other strategy). Give students opportunities to solve the same problem with more than one strategy and discuss which strategy is more efficient and why.

Practice factoring questions

1. Which is the factored form of x^2-6x+7x-42?

(x-7)(x-6)
GCSE Quiz False

(x-7)(x+6)
GCSE Quiz False

(x+7)(x-6)
GCSE Quiz True

(x+7)(x+6)
GCSE Quiz False

Group the terms that have common factors. \left(x^2-6 x\right)+(7 x-42)

 

Factor out the GCF from each binomial.

 

In x^2-6 x, both terms have a factor of x, so x^2-6 x=x(x-6).

 

In 7x-42, both terms have a factor of 7, so 7x-42=7(x-6).

 

So,

 

\begin{aligned} & x^2-6 x+7 x-42 \\\\ & =\left(x^2-6 x\right)+(7 x-42) \\\\ & =x(x-6)+7(x-6) \\\\ & =(x+7)(x-6) \end{aligned}

2. Which is the factored form of 5x^2-45 x+3 x-27?

(5x+3)(x-9)
GCSE Quiz True

(3x-9)(x-3)
GCSE Quiz False

(5x-9)(x+3)
GCSE Quiz False

(x-45)(x+3)
GCSE Quiz False

Group the terms that have common factors. \left(5x^2-45 x\right)+(3 x-27)

 

Factor out the GCF from each binomial.

 

In 5x^2-45 x, both terms have a factor of 5x, so 5x^2-45 x=5 x(x-9).

 

In 3x-27, both terms have a factor of 3, so 3x-27=3(x-9).

 

So,

 

\begin{aligned} & 5 x^2-45 x+3 x-27 \\\\ & =\left(5 x^2-45 x\right)+(3 x-27) \\\\ & =5 x(x-9)+3(x-9) \\\\ & =(5 x+3)(x-9) \end{aligned}

3. Which is the factored form of -2 x^2-4 x+4 x+8?

\left(-2 x^2-4\right)+(4 x+8)
GCSE Quiz False

(x-4)(x+4)
GCSE Quiz False

(2x-4)(x+2)
GCSE Quiz False

(-2x+4)(x+2)
GCSE Quiz True

Group the terms that have common factors. \left(-2 x^2-4x\right)+(4 x+8)

 

Factor out the GCF from each binomial.

 

In -2x^2-4x, both terms have a factor of -2x, so -2x^2-4x=-2x(x+2).

 

In 4x+8, both terms have a factor of 4, so 4x+8=4(x+2).

 

So,

 

\begin{aligned} & -2 x^2-4 x+4 x+8 \\\\& =\left(-2 x^2-4 x\right)+(4 x+8) \\\\& =-2 x(x+2)+4(x+2) \\\\& =(-2 x+4)(x+2)\end{aligned}

4. Which is the factored form of x^2+36 x+\cfrac{1}{3} x+12?

\left(x^2+36 x\right)+\left(\cfrac{1}{3} x+12\right)
GCSE Quiz False

\left(x+\cfrac{1}{3}\right)+\left(x+36\right)
GCSE Quiz True

(x+36)\left(x+\cfrac{1}{3}\right)
GCSE Quiz False

\left(\cfrac{1}{3} x+12\right)(x+6)
GCSE Quiz False

Group the terms that have common factors. \left(x^2+36 x\right)+\left(\cfrac{1}{3} x+12\right)

 

Factor each binomial to find a common factor.

 

In x^2+36x, both terms have a factor of x, so x^2+36 x=x(x+36).

 

In \cfrac{1}{3} x+12, both terms have a factor of \cfrac{1}{3}, so \cfrac{1}{3} x+12=\cfrac{1}{3}(x+36).

 

So,

 

\begin{aligned} & x^2+36 x+\cfrac{1}{3} x+12 \\\\ & =\left(x^2+36 x\right)+\left(\cfrac{1}{3} x+12\right) & =x(x+36)+\cfrac{1}{3}(x+36) \\\\& =\left(x+\cfrac{1}{3}\right)(x+36) \end{aligned}

5. Which is the factored form of x^2+3 x-88?

x^2+(-8 x+11 x)-88
GCSE Quiz False

(x+11)(x-8)
GCSE Quiz True

(x+3)(x-88)
GCSE Quiz False

(x-29)(x+3)
GCSE Quiz False

In order to use factor by grouping, you need to split the middle term. Split it in a way that creates common factors.

 

\begin{aligned} & x^2+3 x-88 \\\\ & =x^2+(-8 x+11 x)-88 \\\\ & =\left(x^2-8 x\right)+(11 x-88) \end{aligned}

 

Factor out the GCF from each binomial.

 

In x^2-8x, both terms have a factor of x, so x^2-8x=x(x-8).

 

In 11x-88, both terms have a factor of 11, so 11x-88=11(x-8).

 

So,

 

\begin{aligned} & x^2+3 x-88 \\\\ & =x^2+(-8 x+11 x)-88 \\\\ & =\left(x^2-8 x\right)+(11 x-88) \\\\ & =x(x-8)+11(x-8) \\\\ & =(x+11)(x-8) \end{aligned}

6. Which is the factored form of x^3+10 x^2-3 x-30?

(x-10)(x+3)
GCSE Quiz False

\left(x^2+10\right)(x-3)
GCSE Quiz False

\left(x^2-3\right)(x+10)
GCSE Quiz True

(10x-1)(x+3)
GCSE Quiz False

Group the terms that have common factors. \left(x^3+10 x^2\right)+(-3 x-30)

 

Factor out the GCF from each binomial.

 

In x^3+10 x^2, both terms have a factor of x^2, so x^3+10 x^2=x^2(x+10).

 

In 3x+30, both terms have a factor of 4, so 3x+30=3(x+10).

 

So,

 

\begin{aligned} & x^3+10 x^2-3 x-30 \\\\ & =\left(x^3+10 x^2\right)-(3 x-30) \\\\ & =x^2(x+10)-3(x+10) \\\\ & =\left(x^2-3\right)(x+10) \end{aligned}

Factor by grouping FAQs

What class in school covers this topic?

This is typically covered in Algebra 1 or Integrated Math II.

Can you factor a monomial?

Yes, for example 8x^2 can be shown as the factors 2x(4x), among other factors.

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