10.6 Introduction to Factoring Polynomials

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.

On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says “multiply” in red. There is an arrow on the bottom pointing to the left that says “factor” in red.

In The Language of Algebra we factored 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 will find the greatest common factor of two numbers.

Example 1

Find the greatest common factor of $24$ and $36 .$

Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor 24 and 36.
Step 2: List all factors--matching common factors in a column.
In each column, circle the common factors.	Circle the 2, 2, and 3 that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.	Bring down the 2, 2, 3 and then multiply.
Step 4: Multiply the factors.		The GCF of 24 and 36 is 12.

Notice that since the GCF is a factor of both numbers, $24$ and $36$ can be written as multiples of $12 .$

$\begin{matrix} 24 = 12 \cdot 2 \\ 36 = 12 \cdot 3 \end{matrix}$

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

Example 3

Find the greatest common factor of $12 x squared$ and $18 x cubed.$

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. Multiply the factors.
	$The GCF of 12 x squared and 18 x cubed is 6 x squared$

Example 4

Find the greatest common factor of $14 x cubed, 8 x squared, 10 x.$

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. Multiply the factors.
	$The GCF of 14 x cubed and 8 x squared, and 10 x is 2 x$

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 $2 \cdot 6 or 3 \cdot 4),$ in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

$\begin{matrix} 2 (x & + & 7) factors \\ 2 \cdot x & + & 2 \cdot 7 \\ 2 x & + & 14 product \end{matrix}$

Here, we will start with a product, like $2 x plus 14,$ and end with its factors, $2 open parenthesis x plus 7 close parenthesis.$ To do this we apply the Distributive Property “in reverse”.

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

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

Example 5

Factor: $2 x plus 14.$

Step 1: Find the GCF of all the terms of the polynomial.	Find the GCF of 2x and 14.
Step 2: Rewrite each term as a product using the GCF.	Rewrite 2x and 14 as products of their GCF, 2. $2 x = 2 \cdot x$ $14 = 2 \cdot 7$
Step 3: Use the Distributive Property 'in reverse' to factor the expression.		$2 open parenthesis x plus 7 close parenthesis$
Step 4: Check by multiplying the factors.		Check:

Notice that in Example 5, we used the word factor as both a noun and a verb:

$Noun, 7 is a factor of 14; Verb, factor 2 from 2 x plus 14$

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

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

In the next example, we factor a variable from a binomial.

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

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

Pay close attention to the signs of the terms in the next example.

Key Concepts

Find the greatest common factor.
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 $a$ , $b$ , $c$ are real numbers, then
  $a (b + c) = a b + a c$ and $a b + a c = a (b + c)$
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 Distributive Property ‘in reverse’ to factor the expression.
4. Check by multiplying the factors.

Section Exercises

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$40, 56$

$45, 75$

$72, 162$

$150, 275$

$3 x, 12$

$4 y, 28$

$10 a, 50$

$5 b, 30$

$16 y, 24 y squared$

$9 x, 15 x squared$

$18 m cubed, 36 m squared$

$12 p to the power 4, 48 p cubed$

12p³

$10 x, 25 x squared, 15 x cubed$

$18 a, 6 a squared, 22 a cubed$

$24 u, 6 u squared, 30 u cubed$

$40 y, 10 y squared, 90 y cubed$

10y

$15 a^{4}, 9 a^{5}, 21 a^{6}$

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

5x³

$27 y squared, 45 y cubed, 9 y to the power 4$

$14 b squared, 35 b cubed, 63 b to the power 4$

7b²

Factor the Greatest Common Factor from a Polynomial

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

$2 x plus 8$

$5 y plus 15$

5(y + 3)

$3 a minus 24$

$4 b minus 20$

4(b − 5)

$9 y minus 9$

$7 x minus 7$

7(x − 1)

$5 m^{2} + 20 m + 35$

$3 n^{2} + 21 n + 12$

3(n² + 7n + 4)

$8 p^{2} + 32 p + 48$

$6 q^{2} + 30 q + 42$

6(q² + 5q + 7)

$8 q squared plus 15 q$

$9 c squared plus 22 c$

c(9c + 22)

$13 k squared plus 5 k$

$17 x squared plus 7 x$

x(17x + 7)

$5 c squared plus 9 c$

$4 q squared plus 7 q$

q(4q + 7)

$5 p squared plus 25 p$

$3 r squared plus 27 r$

3r(r + 9)

$24 q squared minus 12 q$

$30 u squared minus 10 u$

10u(3u − 1)

$y z plus 4 z$

$a b plus 8 b$

b(a + 8)

$60 x minus 6 x cubed$

$55 y - 11 y^{4}$

11y(5 − y³)

$48 r^{4} - 12 r^{3}$

$45 c cubed minus 15 c squared$

15c²(3c − 1)

$4 a cubed minus 4 a b squared$

$6 c cubed minus 6 c d squared$

6c(c² − d²)

$30 u cubed plus 80 u squared$

$48 x cubed plus 72 x squared$

24x²(2x + 3)

$120 y^{6} + 48 y^{4}$

$144 a^{6} + 90 a^{3}$

18a³(8a³ + 5)

$4 q^{2} + 24 q + 28$

$10 y^{2} + 50 y + 40$

10(y² + 5y + 4)

$15 z^{2} - 30 z - 90$

$12 u^{2} - 36 u - 108$

12(u² − 3u − 9)

$3 a^{4} - 24 a^{3} + 18 a^{2}$

$5 p^{4} - 20 p^{3} - 15 p^{2}$

5p²(p² − 4p − 3)

$11 x^{6} + 44 x^{5} - 121 x^{4}$

$8 c^{5} + 40 c^{4} - 56 c^{3}$

8c³(c² + 5c − 7)

$−3 n minus 24$

$−7 p minus 84$

−7(p + 12)

$−15 a squared minus 40 a$

$−18 b squared minus 66 b$

−6b(3b + 11)

$−10 y cubed plus 60 y squared$

$−8 a cubed plus 32 a squared$

−8a²(a − 4)

$−4 u^{5} + 56 u^{3}$

$−9 b^{5} + 63 b^{3}$

−9b³(b² − 7)

Everyday Math

Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of $p$ dollars each is given by the polynomial $−5 p squared plus 150 p.$ Factor the greatest common factor from this polynomial.

Height of a baseball The height of a baseball hit with velocity $80$ feet/second at $4$ feet above ground level is $−16 t^{2} + 80 t + 4,$ with $t =$ the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

−4(4t² − 20t − 1)

Writing Exercises

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

What is the GCF of $y to the power 4$ , $y to the power 5$ , and $y to the power 10$ ? Write a general rule that tells how to find the GCF of $y^{a}$ , $y^{b}$ , and $y^{c}$ .

Answers will vary.

Self Check

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

A self-assessment table for math skills, with columns for 'Confidently,' 'With some help,' and 'No-I don't get it!' The skills listed are finding the greatest common factor and factoring it from a polynomial. — **Figure 10.8**

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

$y^{2} + 8 y - 20$

trinomial

$−6 a to the power 4$

$9 x cubed minus 1$

binomial

$n^{3} - 3 n^{2} + 3 n - 1$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

$16 x^{2} - 40 x - 25$

$5 m plus 9$

$−15$

$y^{2} + 6 y^{3} + 9 y^{4}$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$4 p plus 11 p$

15p

$−8 y cubed minus 5 y cubed$

Add $4 n^{5}, − n^{5}, −6 n^{5}$

−3n⁵

Subtract $10 x squared$ from $3 x squared$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(4 a^{2} + 9 a - 11) + (6 a^{2} - 5 a + 10)$

10a² + 4a − 1

$(8 m^{2} + 12 m - 5) - (2 m^{2} - 7 m - 1)$

$(y^{2} - 3 y + 12) + (5 y^{2} - 9)$

6y² − 3y + 3

$(5 u^{2} + 8 u) - (4 u - 7)$

Find the sum of $8 q cubed minus 27$ and $q^{2} + 6 q - 2$

8q³ + q² + 6q − 29

Find the difference of $x^{2} + 6 x + 8$ and $x^{2} - 8 x + 15$

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

$200 x - \frac{1}{5} x^{2}$ when $x equals 5$

995

$200 x - \frac{1}{5} x^{2}$ when $x equals 0$

$200 x - \frac{1}{5} x^{2}$ when $x equals 15$

2,955

$5 + 40 x - \frac{1}{2} x^{2}$ when $x equals 10$

$5 + 40 x - \frac{1}{2} x^{2}$ when $x equals −4$

−163

$5 + 40 x - \frac{1}{2} x^{2}$ when $x equals 0$

A pair of glasses is dropped off a bridge $640$ feet above a river. The polynomial $−16 t squared plus 640$ gives the height of the glasses $t$ seconds after they were dropped. Find the height of the glasses when $t equals 6.$

64 feet

The fuel efficiency (in miles per gallon) of a bus going at a speed of $x$ miles per hour is given by the polynomial $- \frac{1}{160} x^{2} + \frac{1}{2} x .$ Find the fuel efficiency when $x equals 20$ mph.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

$6 cubed$

216

${(\frac{1}{2})}^{4}$

$open parenthesis −0.5 close parenthesis squared$

0.25

$minus 3 squared$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

$p^{3} \cdot p^{10}$

p¹³

$2 \cdot 2^{6}$

$a \cdot a^{2} \cdot a^{3}$

a⁶

$x \cdot x^{8}$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

$open parenthesis y to the power 4 close parenthesis cubed$

y¹²

${(r^{3})}^{2}$

$open parenthesis 3 squared close parenthesis to the power 5$

3¹⁰

${(a^{10})}^{y}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

$open parenthesis 8 n close parenthesis squared$

64n²

$open parenthesis −5 x close parenthesis cubed$

$open parenthesis 2 a b close parenthesis to the power 8$

256a⁸b⁸

$open parenthesis −10 m n p close parenthesis to the power 4$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

$open parenthesis 3 a to the power 5 close parenthesis cubed$

27a¹⁵

$open parenthesis 4 y close parenthesis squared open parenthesis 8 y close parenthesis$

$open parenthesis x cubed close parenthesis to the power 5 open parenthesis x squared close parenthesis cubed$

x²¹

$open parenthesis 5 s t squared close parenthesis cubed open parenthesis 2 s cubed t to the power 4 close parenthesis squared$

Multiply Monomials

In the following exercises, multiply the monomials.

$open parenthesis −6 p to the power 4 close parenthesis open parenthesis 9 p close parenthesis$

−54p⁵

$(\frac{1}{3} c^{2}) (30 c^{8})$

$(8 x^{2} y^{5}) (7 x y^{6})$

56x³y¹¹

$(\frac{2}{3} m^{3} n^{6}) (\frac{1}{6} m^{4} n^{4})$

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$7 open parenthesis 10 minus x close parenthesis$

70 − 7x

$a^{2} (a^{2} - 9 a - 36)$

$−5 y open parenthesis 125 y cubed minus 1 close parenthesis$

−625y⁴ + 5y

$open parenthesis 4 n minus 5 close parenthesis open parenthesis 2 n cubed close parenthesis$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using various methods.

$(a + 5) (a + 2)$

a² + 7a + 10

$(y - 4) (y + 12)$

$(3 x + 1) (2 x - 7)$

6x² − 19x − 7

$(6 p - 11) (3 p - 10)$

$(n + 8) (n + 1)$

n² + 9n + 8

$(k + 6) (k - 9)$

$(5 u - 3) (u + 8)$

5u² + 37u − 24

$(2 y - 9) (5 y - 7)$

$(p + 4) (p + 7)$

p² + 11p + 28

$(x - 8) (x + 9)$

$(3 c + 1) (9 c - 4)$

27c² − 3c − 4

$(10 a - 1) (3 a - 3)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

$(x + 1) (x^{2} - 3 x - 21)$

x³ − 2x² − 24x − 21

$(5 b - 2) (3 b^{2} + b - 9)$

$(m + 6) (m^{2} - 7 m - 30)$

m³ − m² − 72m − 180

$(4 y - 1) (6 y^{2} - 12 y + 5)$

Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

$\frac{2^{8}}{2^{2}}$

2⁶ or 64

$\frac{a^{6}}{a}$

$\frac{n^{3}}{n^{12}}$

$\frac{1}{n^{9}}$

$\frac{x}{x^{5}}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

$3 to the power 0$

$y to the power 0$

$open parenthesis 14 t close parenthesis to the power 0$

$12 a^{0} - 15 b^{0}$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${(\frac{3}{5})}^{2}$

$\frac{9}{25}$

${(\frac{x}{2})}^{5}$

${(\frac{5 m}{n})}^{3}$

$\frac{125 m^{3}}{n^{3}}$

${(\frac{s}{10 t})}^{2}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(a^{3})}^{2}}{a^{4}}$

a²

$\frac{u^{3}}{u^{2} \cdot u^{4}}$

${(\frac{x}{x^{9}})}^{5}$

$\frac{1}{x^{40}}$

${(\frac{p^{4} \cdot p^{5}}{p^{3}})}^{2}$

$\frac{{(n^{5})}^{3}}{{(n^{2})}^{8}}$

$\frac{1}{n}$

${(\frac{5 s^{2}}{4 t})}^{3}$

Divide Monomials

In the following exercises, divide the monomials.

$72 p^{12} \div 8 p^{3}$

9p⁹

$−26 a^{8} \div (2 a^{2})$

$\frac{45 y^{6}}{−15 y^{10}}$

$- \frac{3}{y^{4}}$

$\frac{−30 x^{8}}{−36 x^{9}}$

$\frac{28 a^{9} b}{7 a^{4} b^{3}}$

$\frac{4 a^{5}}{b^{2}}$

$\frac{11 u^{6} v^{3}}{55 u^{2} v^{8}}$

$\frac{(5 m^{9} n^{3}) (8 m^{3} n^{2})}{(10 m n^{4}) (m^{2} n^{5})}$

$\frac{4 m^{9}}{n^{4}}$

$\frac{42 r^{2} s^{4}}{6 r s^{3}} - \frac{54 r s^{2}}{9 s}$

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

$6 to the power −2$

$\frac{1}{36}$

$open parenthesis −10 close parenthesis to the power −3$

$5 \cdot 2^{−4}$

$\frac{5}{16}$

$open parenthesis 8 n close parenthesis to the power −1$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$x^{−3} \cdot x^{9}$

x⁶

$r^{−5} \cdot r^{−4}$

$(u v^{−3}) (u^{−4} v^{−2})$

$\frac{1}{u^{3} v^{5}}$

${(m^{5})}^{−1}$

${(k^{−2})}^{−3}$

k⁶

$\frac{q^{4}}{q^{20}}$

$\frac{b^{8}}{b^{−2}}$

b¹⁰

$\frac{n^{−3}}{n^{−5}}$

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

$5,300,000$

5.3 × 10⁶

$0.00814$

The thickness of a piece of paper is about $0.097$ millimeter.

9.7 × 10⁻² millimeter

According to www.cleanair.com, U.S. businesses use about $21,000,000$ tons of paper per year.

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$2.9 \times 10^{4}$

29,000

$1.5 \times 10^{8}$

$3.75 \times 10^{−1}$

0.375

$9.413 \times 10^{−5}$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

$(3 \times 10^{7}) (2 \times 10^{−4})$

6,000

$(1.5 \times 10^{−3}) (4.8 \times 10^{−1})$

$\frac{6 \times 10^{9}}{2 \times 10^{−1}}$

30,000,000,000

$\frac{9 \times 10^{−3}}{1 \times 10^{−6}}$

Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$5 n, 45$

$8 a, 72$

$12 x squared, 20 x cubed, 36 x to the power 4$

4x²

$9 y^{4}, 21 y^{5}, 15 y^{6}$

Factor the Greatest Common Factor from a Polynomial

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

$16 u minus 24$

8(2u − 3)

$15 r plus 35$

$6 p squared plus 6 p$

6p(p + 1)

$10 c squared minus 10 c$

$−9 a^{5} - 9 a^{3}$

−9a³(a² + 1)

$−7 x^{8} - 28 x^{3}$

$5 y^{2} - 55 y + 45$

5(y² − 11y + 9)

$2 q^{5} - 16 q^{3} + 30 q^{2}$

Chapter Practice Test

For the polynomial $8 y^{4} - 3 y^{2} + 1$

ⓐ Is it a monomial, binomial, or trinomial?
ⓑ What is its degree?

ⓐ trinomial
ⓑ 4

In the following exercises, simplify each expression.

$(5 a^{2} + 2 a - 12) + (9 a^{2} + 8 a - 4)$

$(10 x^{2} - 3 x + 5) - (4 x^{2} - 6)$

6x² − 3x + 11

${(- \frac{3}{4})}^{3}$

$n \cdot n^{4}$

n⁵

$open parenthesis 10 p cubed q to the power 5 close parenthesis squared$

$(8 x y^{3}) (−6 x^{4} y^{6})$

−48x⁵y⁹

$4 u (u^{2} - 9 u + 1)$

$(s + 8) (s + 9)$

s² + 17s + 72

$(m + 3) (7 m - 2)$

$(11 a - 6) (5 a - 1)$

55a² − 41a + 6

$(n - 8) (n^{2} - 4 n + 11)$

$(4 a + 9 b) (6 a - 5 b)$

24a² + 34ab − 45b²

$\frac{5^{6}}{5^{8}}$

${(\frac{x^{3} \cdot x^{9}}{x^{5}})}^{2}$

x¹⁴

${(47 a^{18} b^{23} c^{5})}^{0}$

$\frac{24 r^{3} s}{6 r^{2} s^{7}}$

$\frac{4 r}{s^{6}}$

$\frac{8 y^{2} - 16 y + 20}{4 y}$

$(15 x y^{3} - 35 x^{2} y) \div 5 x y$

3y² − 7x

$4 to the power −1$

$open parenthesis 2 y close parenthesis to the power −3$

$\frac{1}{8 y^{3}}$

$p^{−3} \cdot p^{−8}$

$\frac{x^{4}}{x^{−5}}$

x⁹

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

$80 a^{3} + 120 a^{2} + 40 a$

$−6 x squared minus 30 x$

−6x(x + 5)

According to www.cleanair.org, the amount of trash generated in the US in one year averages out to $112,000$ pounds of trash per person. Write this number in scientific notation.

Convert $5.25 \times 10^{−4}$ to decimal form.

0.000525

In the following exercises, simplify, and write your answer in decimal form.

$(2.4 \times 10^{8}) (2 \times 10^{−5})$

$\frac{9 \times 10^{4}}{3 \times 10^{−1}}$

$300,000$

A hiker drops a pebble from a bridge $240$ feet above a canyon. The polynomial $−16 t squared plus 240$ gives the height of the pebble $t$ seconds a after it was dropped. Find the height when $t equals 3.$

	$6 x squared plus 5 x$
Find the GCF of $6 x squared$ and $5 x$ and the math that goes with it.
Rewrite each term as a product.
Factor the GCF.	$x open parenthesis 6 x plus 5 close parenthesis$
Check by multiplying.
$x open parenthesis 6 x plus 5 close parenthesis$ $x \cdot 6 x + x \cdot 5$ $6 x squared plus 5 x ✓$

	$14 x^{3} + 8 x^{2} - 10 x$
Rewrite each term using the GCF, 2x.
Factor the GCF.	$2 x (7 x^{2} + 4 x - 5)$



Rewrite each term as a product using the GCF.
Use the Distributive Property 'in reverse' to factor the GCF.
Check by multiplying the factors to get the original polynomial.



Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying the factors.