7.1 Greatest Common Factor and Factor by Grouping

Learning objectives

Learning objectives

• Find the greatest common factor of two or more expressions

• Factor the greatest common factor from a polynomial

• Factor by grouping

Find the Greatest Common Factor of Two or More Expressions

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

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.

First we’ll find the GCF of two numbers.

Example 7.1

Solution

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

We summarize the steps we use to find the GCF below.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example 7.2

Solution

Example 7.3

Solution

Example 7.4

Solution

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as $2\cdot{}6\,\mathrm{or}\,3\cdot{}4),$ in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

Now we will start with a product, like $2x+14$, and end with its factors, $2(x+7)$. 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.”

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 7.5

Example 7.6

Solution

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example 7.7

Solution

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example 7.8

Solution

We start by finding the GCF of all three terms.

Example 7.9

Solution

Example 7.10

Solution

In a previous example we found the GCF of $21{x}^{3},9{x}^{2},15x$ to be $3x$.

Example 7.11

Solution

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

Example 7.12

Solution

When the leading coefficient is negative, the GCF will be negative.

Example 7.13

Solution

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

Example 7.14

Solution

The GCF is the binomial $q+7$.

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating 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 7.15

Example 7.16

Solution

$\begin{array}{lccc} \mathrm{There is no GCF in all four terms.} & & & \,{x}^{2}+3x\,-2x-6 \\ \mathrm{Separate into two parts.} & & & \,\underset{\mathrm{⎵}}{{x}^{2}+3x}\,\underset{\mathrm{⎵}}{-2x-6} \\ \begin{array}{l} \mathrm{Factor the GCF from both parts. Be careful} \\ \mathrm{with the signs when factoring the GCF from} \\ \mathrm{the last two terms.} \\ \end{array} & & & \,\begin{array}{c} x(x+3)-2(x+3) \\ (x+3)(x-2) \\ \end{array} \\ \mathrm{Check on your own by multiplying.} & & & \\ \end{array}$

Section 7.1 Exercises

Section 7.1 Exercises

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

Factor by Grouping

In the following exercises, factor by grouping.

Mixed Practice

In the following exercises, factor.

Everyday Math

Writing Exercises

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

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

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1. Find the GCF: $12{x}^{2},18{x}^{3}$.

2. Find the GCF: $16{y}^{2},24{y}^{3}$.

3. Find the GCF: $6a{b}^{4},8{a}^{2}b$.

4. Find the GCF: $9{m}^{5}{n}^{2},12{m}^{3}n$.

1. Find the greatest common factor: $25{m}^{4},35{m}^{3},20{m}^{2}$.

2. Find the greatest common factor: $14{x}^{3},70{x}^{2},105x$.

3. Factor: $6a+24$.

4. Factor: $2b+14$.

1. Factor: $14x+14$.

2. Factor: $12p+12$.

3. Factor: $18u-36$.

4. Factor: $30y-60$.

1. Factor: $5{x}^{2}-25x+15$.

2. Factor: $3{y}^{2}-12y+27$.

3. Factor: $2{x}^{3}+12{x}^{2}$.

4. Factor: $6{y}^{3}-15{y}^{2}$.

1. Factor: $20{x}^{3}-10{x}^{2}+14x$.

2. Factor: $24{y}^{3}-12{y}^{2}-20y$.

3. Factor: $9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}$.

4. Factor: $3{p}^{3}-6{p}^{2}q+9p{q}^{3}$.

1. Factor: $-16z-64$.

2. Factor: $-9y-27$.

3. Factor: $-4{b}^{2}+16b$.

4. Factor: $-7{a}^{2}+21a$.

1. Factor: $4m(m+3)-7(m+3)$.

2. Factor: $8n(n-4)+5(n-4)$.

3. Factor: $xy+8y+3x+24$.

4. Factor: $ab+7b+8a+56$.

1. Factor: ${x}^{2}+2x-5x-10$.

2. Factor: ${y}^{2}+4y-7y-28$.

3. $3x,10{x}^{2}$

4. $8{w}^{2},24{w}^{3}$

1. $10{p}^{3}q,12p{q}^{2}$

2. $12{m}^{2}{n}^{3},30{m}^{5}{n}^{3}$

3. $10{a}^{3},12{a}^{2},14a$

4. $35{x}^{3},10{x}^{4},5{x}^{5}$

1. $4x+20$

2. $6m+9$

3. $9q+9$

4. $8m-8$

1. $9n-63$

2. $3{x}^{2}+6x-9$

3. $8{p}^{2}+4p+2$

4. $8{y}^{3}+16{y}^{2}$

1. $5{x}^{3}-15{x}^{2}+20x$

2. $12x{y}^{2}+18{x}^{2}{y}^{2}-30{y}^{3}$

3. $-2x-4$

4. $5x(x+1)+3(x+1)$

1. $3b(b-2)-13(b-2)$

2. $xy+2y+3x+6$

3. $uv-9u+2v-18$

4. ${b}^{2}+5b-4b-20$

1. ${p}^{2}+4p-9p-36$

2. $-20x-10$

3. $3{x}^{3}-7{x}^{2}+6x-14$

4. ${x}^{2}+xy+5x+5y$

1. Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression ${w}^{2}-6w$, where $w=$ width. Factor the greatest common factor from the polynomial.

2. The greatest common factor of 36 and 60 is 12. Explain what this means.