1.1 Greatest Common Factor and Factor by Grouping

Learning Objectives

By the end of this section, you will be able to:

  • Find the greatest common factor of two or more expressions
  • Factor the greatest common factor from a polynomial
  • Factor by grouping

Try It

Before you get started, take this readiness quiz:

1) Factor [latex]56[/latex] into primes.
2) Find the least common multiple (LCM) of [latex]18[/latex] and [latex]24[/latex].
3) Multiply: [latex](-3a\left(7a+8b\right)[/latex].

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

8 times 7 is 56. Here 8 and 7 are factors and 56 is the product. An arrow pointing from 8 times 7 to 56 is labeled multiply. An arrow pointing from 56 to 8 times 7 is labeled factor. 2x open parentheses x plus 3 close parentheses equals 2x squared plus 6x. Here the left side of the equation is labeled factors and the right side is labeled products.
Figure 1.1.1

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

We summarize the steps we use to find the greatest common factor.

HOW TO

Find the greatest common factor (GCF) of two expressions.

  1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
  2. List all factors—matching common factors in a column. In each column, circle the common factors.
  3. Bring down the common factors that all expressions share.
  4. Multiply the factors.

The next example will show us the steps to find the greatest common factor of three expressions.

Example 1.1.1

Find the greatest common factor of [latex]21x^3,\;9x^2,\;15x[/latex].

Solution

Step 1: Factor each coefficient into primes and write the variables with exponents in expanded form.

Circle the common factors in each column.

Bring down the common factors.

.
Figure 1.1.2

Step 2: Multiply the factors.

[latex]GCF=\;3x[/latex]

Step 3: Worded statement.

The GCF of [latex]21{x}^{3}[/latex], [latex]9{x}^{2}[/latex] and [latex]15x[/latex] is [latex]3x[/latex].

Try It

4) Find the greatest common factor: [latex]25{m}^{4},35{m}^{3},20{m}^{2}.[/latex]

Solution

[latex]5{m}^{2}[/latex]

Try It

5) Find the greatest common factor: [latex]14{x}^{3},70{x}^{2},105x.[/latex]

Solution

[latex]7x[/latex]

Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, [latex]12[/latex] as [latex]2·6[/latex] or [latex]3·4.[/latex] In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as [latex]3{x}^{2}+15x,[/latex] and end with its factors, [latex]3x\left(x+5\right).[/latex] To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive Property

If [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers, then

[latex]a\left(b+c\right)=ab+ac\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}ab+ac=a\left(b+c\right)[/latex]

 

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 1.1.2

How to Use the Distributive Property to factor a polynomials. Factor: [latex]8{m}^{3}-12{m}^{2}n+20m{n}^{2}.[/latex]

Solution
Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.
Figure 1.1.3
Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.
Figure 1.1.4
In step 3, use the reverse Distributive Property to factor the expression as 4m open parentheses 2 m squared minus 3 mn plus 5 n squared close parentheses.
Figure 1.1.5
Step 4 is to check by multiplying the factors. By multiplying the factors, we get the original polynomial.
Figure 1.1.6

Try It

6) Factor: [latex]9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}.[/latex]

Solution

[latex]3{y}^{2}\left(3x+2{x}^{2}+7y\right)[/latex]

Try It

7) Factor: [latex]3{p}^{3}-6{p}^{2}q+9p{q}^{3}.[/latex]

Solution

[latex]3p\left({p}^{2}-2pq+3{q}^{3}\right)[/latex]

Factor the greatest common factor from a polynomial.

  1. Find the GCF of all the terms of the polynomial.
  2. Rewrite each term as a product using the GCF.
  3. Use the “reverse” Distributive Property to factor the expression.
  4. Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb:

[latex]\begin{array}{cccc}\text{Noun:}\hfill & & & \phantom{\rule{8em}{0ex}}\text{7 is a}\phantom{\rule{0.2em}{0ex}}{\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{of 14}\hfill \\ \text{Verb:}\hfill & & & \phantom{\rule{8em}{0ex}}{\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{3 from}\phantom{\rule{0.2em}{0ex}}3a+3\hfill \end{array}[/latex]

Example 1.1.3

Factor: [latex]5x^3\;-\;25x^2[/latex].

Solution

Step 1: Find the GCF of [latex]5{x}^{3}[/latex] and [latex]25{x}^{2}[/latex].

.
Figure 1.1.7

[latex]\begin{eqnarray*}\text{GCF}\;\;\;\;\;&=&\;5x^2\\&=&5x^3\;-\;25x^2\\\text{Rewrite each term.}\;\;\;\;\;&=&{\color{red}{5}}{\color{red}{x}}^{\color{red}{2}}\;\times\;x\;-\;{\color{red}{5}}{\color{red}{x}}^{\color{red}{2}}\;\times5\\\text{Factor the GCF.}\;\;\;\;\;&=&5x^2(x-5)\end{eqnarray*}[/latex]

Step 2: Check.

[latex]\begin{eqnarray*}&=&5x^2(x-5)\\&=&5x^2\times x-5x^2\times5\\&=&5x^3-25x^3\checkmark\end{eqnarray*}[/latex]

Try It

8) Factor: [latex]2{x}^{3}+12{x}^{2}.[/latex]

Solution

[latex]2{x}^{2}\left(x+6\right)[/latex]

Try It

9) Factor: [latex]6{y}^{3}-15{y}^{2}.[/latex]

Solution

[latex]3{y}^{2}\left(2y-5\right)[/latex]

Example 1.1.4

Factor: [latex]8x^3y\text{,}-10x^2y^2\;\text{and }12xy^3[/latex].

Solution

Step 1: Find the GCF.

.
Figure 1.1.8

[latex]\begin{eqnarray*}\text{GCF}\;\;\;\;\;&=&2xy\\&=&8x^3-10x^2y^2+12xy^3\\\text{Rewrite using GCF.}\;\;\;\;\;&=&{\color{red}{2}}{\color{red}{x}}{\color{red}{y}}\times4x^2-{\color{red}{2}}{\color{red}{x}}{\color{red}{y}}\times5xy+{\color{red}{2}}{\color{red}{x}}{\color{red}{y}}\times6y^2\\\text{Factor the GCF.}\;\;\;\;\;&=&2xy(4x^2-5xy+6y^2)\end{eqnarray*}[/latex]

Step 2: Check.

[latex]\begin{eqnarray*}&=&2xy(4x^2-5xy+6y^2)\\&=&2xy\times4x^2-2xy\times5xy+2xy\times6y^2\\&=&8x^3y-10x^2y^2+12xy^3\checkmark\end{eqnarray*}[/latex]

Try It

10) Factor: [latex]15{x}^{3}y-3{x}^{2}{y}^{2}+6x{y}^{3}.[/latex]

Solution

[latex]3xy\left(5{x}^{2}-xy+2{y}^{2}\right)[/latex]

Try It

11) Factor: [latex]8{a}^{3}b+2{a}^{2}{b}^{2}-6a{b}^{3}.[/latex]

Solution

[latex]2ab\left(4{a}^{2}+ab-3{b}^{2}\right)[/latex]

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 1.1.5

Factor: [latex]-4a^3+36a^2-8a[/latex].

Solution

Step 1: Rewrite each term using the GCF, [latex]-4a[/latex].

The leading coefficient is negative, so the GCF will be negative.

[latex]\begin{eqnarray*}&=&{\color{red}{-}}{\color{red}{4}}{\color{red}{a}}\times a^2-{\color{red}{\left(-4a\right)}}\times9a+{\color{red}{\left(-4a\right)}}\times2\\\text{Factor the GCF.}\;\;\;\;\;&=&-4a(a^2-9a+2)\end{eqnarray*}[/latex]

Step 2: Check.

[latex]\begin{eqnarray*}&=&-4a(a^2-9a+2)\\&=&-4a\times a^2-(-4a)\times9a+(-4a)\times2\\&=&-4a^3+36a^2-8a\checkmark\end{eqnarray*}[/latex]

Try It

12) Factor: [latex]-4{b}^{3}+16{b}^{2}-8b.[/latex]

Solution

[latex]-4b\left({b}^{2}-4b+2\right)[/latex]

Try It

13) Factor: [latex]-7{a}^{3}+21{a}^{2}-14a.[/latex]

Solution

[latex]-7a\left({a}^{2}-3a+2\right)[/latex]

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 1.1.6

Factor: [latex]3y(y+7)-4(y+7)[/latex].

Solution

Step 1: Factor the GCF.

The GCF is the binomial [latex]y+7[/latex].

[latex]\begin{eqnarray*}&=&3y{\color{red}{(}}{\color{red}{y}}{\color{red}{+}}{\color{red}{7}}{\color{red}{)}}-4{\color{red}{(}}{\color{red}{y}}{\color{red}{+}}{\color{red}{7}}{\color{red}{)}}\\{\color{black}{\text{Factor the GCF.}}}\;\;\;\;\;{\color{black}{}}&=&{\color{black}{(}}{\color{black}{y}}{\color{black}{+}}{\color{black}{7}}{\color{black}{)}}{\color{black}{(}}{\color{black}{y}}{\color{black}{-}}{\color{black}{4}}{\color{black}{)}}\end{eqnarray*}[/latex]

Step 2: Check on your own by multiplying.

Try It

14) Factor: [latex]4m\left(m+3\right)-7\left(m+3\right).[/latex]

Solution

[latex]\left(m+3\right)\left(4m-7\right)[/latex]

Try It

15) Factor: [latex]8n\left(n-4\right)+5\left(n-4\right).[/latex]

Solution

[latex]\left(n-4\right)\left(8n+5\right)[/latex]

Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.

Example 1.1.7

How to Factor a Polynomial by Grouping. Factor by grouping: [latex]xy+3y+2x+6[/latex].

Solution
Step 1 is to group the terms with common factors. There is no greatest common factor in all the four terms of xy plus 3y plus 2x plus 6. So, separate the first two terms from the second two.
Figure 1.1.9
Step 2 is to factor out the common factor in each group. By factoring the GCF from the first 2 terms, we get y open parentheses x plus 3 close parentheses plus 2x plus 6. Factoring the GCF from the second 2 terms, we get y open parentheses x plus 3 close parentheses plus 2 open parentheses x plus 3 close parentheses.
Figure 1.1.10
Step 3 is to factor the common factor from the expression. Notice that each term has a common factor of x plus 3. By factoring this out, we get open parentheses x plus 3 close parentheses open parentheses y plus 2 close parentheses
Figure 1.1.11
Step 4 is to check by multiplying the expressions to get the result xy plus 3y plus 2x plus 6.
Figure 1.1.12

Try It

16) Factor by grouping: [latex]xy+8y+3x+24.[/latex]

Solution

[latex]\left(x+8\right)\left(y+3\right)[/latex]

Try It

17) Factor by grouping: [latex]ab+7b+8a+56.[/latex]

Solution

[latex]\left(a+7\right)\left(b+8\right)[/latex]

Factor by grouping.

  1. Group terms with common factors.
  2. Factor out the common factor in each group.
  3. Factor the common factor from the expression.
  4. Check by multiplying the factors.

Example 1.1.8

Factor by Grouping:
a. [latex]x^2+3x-2x-6[/latex]
b. [latex]6x^2-3x-4x+2[/latex]
Solution

a.

Step 1: Find the GCF. 

There is no GCF for all four terms, separate into two parts.

[latex]{\color{black}{x}}{^{\color{black}{2}}{\color{black}{+}}{\color{black}{3}}{\color{black}{x}}{\color{black}{-}}{\color{black}{2}}{\color{black}{x}}{\color{black}{-}}{\color{black}{6}}{\;=\;{\color{red}{x}}^{\color{red}{2}}{\color{red}{+}}{\color{red}{3}}{\color{red}{x}}\;\text{and}\;{\color{blue}{-}}{\color{blue}{2}}{\color{blue}{x}}{\color{blue}{-}}}{\color{blue}{6}}{}}[/latex]

Step 2: Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.

[latex]\begin{eqnarray*}&=&{\color{blue}{x}}(x+3){\color{blue}{-}}{\color{blue}{2}}(x+3)\\\text{Factor out common factor.}&=&(x+3)({\color{blue}{x}}{\color{blue}{-}}{\color{blue}{2}})\end{eqnarray*}[/latex]

Step 3: Check on your own by multiplying. 

b.

Step 1: Find the GCF. 

There is no GCF for all four terms, separate into two parts.

[latex]{\color{black}{6}}{{\color{black}{x}}^{\color{black}{2}}{\color{black}{-}}{\color{black}{3}}{\color{black}{x}}{\color{black}{-}}{\color{black}{4}}{\color{black}{x}}{\color{black}{+}}{\color{black}{2}}{\color{black}{\;=\;}}{\color{red}{6}}{\color{red}{x}}^{\color{red}{2}}{\color{red}{-}}{\color{red}{3}}{\color{red}{x}}{\color{blue}{\;}}{\color{black}{\text{and}}}{\color{blue}{\;}}{\color{blue}{-}}{\color{blue}{4}}{\color{blue}{x}}{\color{blue}{+}}}{\color{blue}{2}}[/latex]

Step 2: Factor the GCF from both parts.

[latex]\begin{eqnarray*}&=&{\color{blue}{3}}{\color{blue}{x}}(2x-1){\color{blue}{-}}{\color{blue}{2}}(2x-1)\\\text{Factor out common factor.}&=&(2x-1)({\color{blue}{3}}{\color{blue}{x}}{\color{blue}{-}}{\color{blue}{2}})\end{eqnarray*}[/latex]

Step 3: Check on your own by multiplying.

Try It

18) Factor by grouping:

a. [latex]{x}^{2}+2x-5x-10[/latex]
b. [latex]20{x}^{2}-16x-15x+12.[/latex]

Solution

a. [latex]\left(x-5\right)\left(x+2\right)[/latex]
b. [latex]\left(5x-4\right)\left(4x-3\right)[/latex]

Try It

19) Factor by grouping:

a. [latex]{y}^{2}+4y-7y-28[/latex]
b. [latex]42{m}^{2}-18m-35m+15.[/latex]

Solution

a. [latex]\left(y+4\right)\left(y-7\right)[/latex]
b. [latex]\left(7m-3\right)\left(6m-5\right)[/latex]

Key Concepts

  • How to find the greatest common factor (GCF) of two expressions.
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors.
  • Distributive Property: If [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers, then
    [latex]a\left(b+c\right)=ab+ac\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}ab+ac=a\left(b+c\right)[/latex]

    The form on the left is used to multiply. The form on the right is used to factor.

  • How to factor the greatest common factor from a polynomial.
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the “reverse” Distributive Property to factor the expression.
    4. Check by multiplying the factors.
  • Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

    [latex]\begin{array}{cccc}\text{Noun:}\hfill & & & \text{7 is a}\phantom{\rule{0.2em}{0ex}}{\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{of 14}\hfill \\ \text{Verb:}\hfill & & & {\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{3 from}\phantom{\rule{0.2em}{0ex}}3a+3\hfill \end{array}[/latex]

  • How to factor by grouping.
    1. Group terms with common factors.
    2. Factor out the common factor in each group.
    3. Factor the common factor from the expression.
    4. Check by multiplying the factors.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Glossary

factoring
Splitting a product into factors is called factoring.
greatest common factor
The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
definition

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