Topics
1. Polynomials and Factoring
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United States of America
Grades 9-12

1.06 Factoring by grouping

Lesson
Practice

Introduction

Factoring by grouping is an application of factoring a greatest common factor, as we learned in lesson  9.03 Factoring GCF  . We will use this concept in this lesson to factor different types of polynomial expressions.

Factoring by grouping

Exploration

Consider the polynomial expression x^4+7x^3+6x+42.

Factor out the greatest common factor for each of the following:

  • x^4+7x^3
  • 6x+42

1. Based on your answers above, what do you think the factored form of x^4+7x^3+6x+42 could be?

When a polynomial with four terms does not have a GCF, sometimes we can group pairs of terms and factor their GCFs to factor the polynomial.

Recall the expansion of the following expression:(a+b)(x+y)=a(x+y)+b(x+y)= ax + ay + bx + byWe can reverse these steps with expressions that contain four terms by factoring in pairs. This is called factoring by grouping.

Factoring by grouping

A method for factoring an expression containing at least four terms, by grouping the terms in pairs and taking out common factors

Follow the steps shown with the example to factor by grouping:

A figure showing the steps in factoring the expression 6 x cubed minus 16 plus 4 x minus 24 x squared. Speak to your teacher for more information.

Examples

Example 1

Factor the expression 10 xy + 4x + 15y + 6.

Worked Solution
Create a strategy

We arrange the terms first, grouping those with common factors. We factor out the GCF on each pair and the common binomial factor afterward.

Apply the idea
\displaystyle 10 xy + 4x + 15y + 6\displaystyle =\displaystyle \left(10 xy + 4x\right)+ \left(15y + 6\right)Group based on common factors
\displaystyle =\displaystyle 2x\left(5y + 2\right)+ 3\left(5y + 2\right)Factor out each GCF \left(2x \text{ and } 3 \right)
\displaystyle =\displaystyle \left(5y + 2\right)\left(2x + 3\right)Factor out the common binomial factor

Since \left(5y + 2\right)\left(2x + 3\right) cannot be factored further, it is the final answer.

Reflect and check

Alternatively, we can group 10xy \text{ and } 15y and 4x \text{ and } 6 together and get the same answer.

\displaystyle 10 xy + 4x + 15y + 6\displaystyle =\displaystyle \left(10 xy + 15y\right)+ \left(4x + 6\right)Group based on common factors
\displaystyle =\displaystyle 5y\left(2x + 3\right)+ 2\left(2x + 3\right)Factor out each GCF \left(5y \text{ and } 2 \right)
\displaystyle =\displaystyle \left(2x + 3\right)\left(5y + 2\right)Factor out the common binomial factor

We can perform a midway check that we are factoring by grouping appropriately when we factor out the GCF from each set of binomials in the step 2x\left(5y + 2\right)+ 3\left(5y + 2\right).

If we factor out the GCF at this step and the binomial factors are not equivalent, then we will want to check that we factored out the GCF correctly. If the factoring is correct, we may need to try a different approach. There may be a better way to arrange the terms from the polynomial.

Not every polynomial expression will be factorable, but we can try a few different approaches, checking our work along the way.

Example 2

Show at least two different ways we can arrange and group the polynomial 4a^2-10b+5ab-8a and factor it.

Worked Solution
Create a strategy

Determine if the polynomial has common factors between the first and second pair of terms, then factor it and rearrange the polynomial so that the first or second set of terms has a common factor, then factor it again.

Apply the idea

Write the expression as 4a^2+5ab-8a-10b and factor it.

\displaystyle 4a^2-10b+5ab-8a\displaystyle =\displaystyle 4a^{2} + 5ab - 8a - 10b Rearrange the terms
\displaystyle 4a^{2} + 5ab - 8a - 10b\displaystyle =\displaystyle \left(4a^{2} + 5ab \right)+ \left(- 8a - 10b\right)Group based on common factors
\displaystyle =\displaystyle a\left(4a+5b\right) - 2\left(4a+5b\right)Factor out the GCF \left(a \text{ and } -2 \right)
\displaystyle =\displaystyle \left(4a + 5b\right)\left(a - 2\right)Factor out the common binomial factor

Since \left(4a + 5b\right)\left(a - 2\right) cannot be factored further, it is the final answer.

Write the expression as 4a^2-8a-10b+5ab and factor it.

\displaystyle 4a^2-10b+5ab-8a\displaystyle =\displaystyle 4a^2-8a-10b+5ab Rearrange the terms
\displaystyle 4a^2-8a-10b+5ab\displaystyle =\displaystyle (4a^2-8a)+(-10b+5ab)Group based on common factors
\displaystyle =\displaystyle 4a(a-2)+5b(-2+a)Factor out the GCF \left(4a \text{ and } 5b \right)
\displaystyle =\displaystyle (a-2)(4a+5b)Factor out the common binomial factor

Since (a-2)(4a+5b) cannot be factored further, it is the final answer.

Reflect and check

Alternatively, we can group 4a^{2} \text{ and } - 8a and 5ab \text{ and } - 10b together and get the same answer.

\displaystyle 4a^{2} + 5ab - 8a - 10b\displaystyle =\displaystyle \left(4a^{2} - 8a \right)+ \left(5ab - 10b\right)Group based on common factors
\displaystyle =\displaystyle 4a\left(a- 2\right) + 5b\left(a-2\right)Factor out the GCF \left(4a \text{ and } 5b \right)
\displaystyle =\displaystyle \left(a - 2\right)\left(4a + 5b\right)Factor out the common binomial factor

We can check the answer by multiplying the factored form \left(4a + 5b\right)\left(a - 2\right).

\displaystyle \left(4a + 5b\right)\left(a - 2\right)\displaystyle =\displaystyle a\left(4a + 5b\right) -2\left(4a + 5b\right)Distributive property
\displaystyle =\displaystyle 4a^{2} + 5ab - 8a- 10bDistributive property

Example 3

Factor the expression 12x^3-4x^2-3x+1.

Worked Solution
Create a strategy

The first two terms have a common factor, and the only common factor of the second two terms is 1. We will begin our factoring without rearranging the terms.

Apply the idea
\displaystyle 12x^3-4x^2-3x+1\displaystyle =\displaystyle (12x^3-4x^2)+(-3x+1)Group based on common factors
\displaystyle =\displaystyle 4x^2(3x-1) -1(3x-1)Factor out each GCF \left(4x^2 \text{ and } -1 \right)
\displaystyle =\displaystyle (3x-1)(4x^2-1)Factor out the common binomial factor

Since (3x-1)(4x^2-1) has a special product 4x^2-1, we can factor it using the product of a sum and difference rule \left(a+b\right)\left(a-b\right) = a^{2} - b^{2} So, 4x^2-1=(2x+1)(2x-1)

The fully factored form of 12x^3-4x^2-3x+1 is (3x-1)(2x+1)(2x-1).

Reflect and check

Note that the sign of 3x and 1 in the expression 12x^3-4x^2-3x+1 change when the common factor -1 was factored out. The common binomial factors would not have been equivalent if -1 was not factored out of the binomial expression.

Idea summary

Follow these steps when factoring by grouping:

  1. Factor out the GCF from the expression, if possible
  2. Arrange the terms so that the first two have a common factor and the last two have a common factor, if possible
  3. Factor out the GCF for each pair of terms
  4. Factor out the common binomial expression
  5. Confirm that the binomial factors cannot be factored further, otherwise continue factoring

Outcomes

A.SSE.A.2

Use the structure of an expression to identify ways to rewrite it.

A.SSE.B.3

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

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