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.

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

Solution

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.

Find the greatest common factor of $5 x and 15 .$

Solution

Factor each number into primes. Circle the common factors in each column. Bring down the common factors.
	The GCF of 5x and 15 is 5.

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.

Find the greatest common factor of $12 x^{2}$ and $18 x^{3} .$

Solution

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.
	$The GCF of 12 x^{2} and 18 x^{3} is 6 x^{2}$

Find the greatest common factor of $14 x^{3}, 8 x^{2}, 10 x .$

Solution

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.
	$The GCF of 14 x^{3} and 8 x^{2}, and 10 x is 2 x$

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 $2 x + 14,$ and end with its factors, $2 (x + 7) .$ 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!

Factor: $2 x + 14 .$

Solution

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.		$2 (x + 7)$
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:

\begin{matrix} Noun & 7 is a factor of 14 \\ Verb & factor 2 from 2 x + 14 \end{matrix}

Factor: $3 a + 3 .$

Solution



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.

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

Factor: $12 x - 60 .$

Solution



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

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

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

Solution



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

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

Factor: $6 x^{2} + 5 x .$

Solution

	$6 x^{2} + 5 x$
Find the GCF of $6 x^{2}$ and $5 x$ and the math that goes with it.
Rewrite each term as a product.
Factor the GCF.	$x (6 x + 5)$
Check by multiplying.
$x (6 x + 5)$ $x \cdot 6 x + x \cdot 5$ $6 x^{2} + 5 x ✓$

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

Factor: $4 x^{3} - 20 x^{2} .$

Solution



Rewrite each term.
Factor the GCF.
Check.

Factor: $21 y^{2} + 35 y .$

Solution

Find the GCF of $21 y^{2}$ and $35 y$

Rewrite each term.
Factor the GCF.

Factor: $14 x^{3} + 8 x^{2} - 10 x .$

Solution

Previously, we found the GCF of $14 x^{3}, 8 x^{2}, and 10 x$ to be $2 x .$

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

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

Factor: $−9 y - 27 .$

Solution

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y and 27 is 9.
Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF.
	$- 9 y - 27$
Rewrite each term using the GCF.
Factor the GCF.	$- 9 (y + 3)$

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

Factor: $−4 a^{2} + 16 a .$

Solution

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

Since the leading coefficient is negative, the GCF is negative, −4a.
	$−4 a^{2} + 16 a$
Rewrite each term.
Factor the GCF.	$- 4 a (a - 4)$
Check on your own by multiplying.

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.

$40, 56$

$45, 75$

Solution

$72, 162$

$150, 275$

Solution

$3 x, 12$

$4 y, 28$

Solution

$10 a, 50$

$5 b, 30$

Solution

$16 y, 24 y^{2}$

$9 x, 15 x^{2}$

Solution

$18 m^{3}, 36 m^{2}$

$12 p^{4}, 48 p^{3}$

Solution

12p³

$10 x, 25 x^{2}, 15 x^{3}$

$18 a, 6 a^{2}, 22 a^{3}$

Solution

$24 u, 6 u^{2}, 30 u^{3}$

$40 y, 10 y^{2}, 90 y^{3}$

Solution

10y

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

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

Solution

5x³

$27 y^{2}, 45 y^{3}, 9 y^{4}$

$14 b^{2}, 35 b^{3}, 63 b^{4}$

Solution

7b²

Factor the Greatest Common Factor from a Polynomial

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

$2 x + 8$

$5 y + 15$

Solution

5(y + 3)

$3 a - 24$

$4 b - 20$

Solution

4(b − 5)

$9 y - 9$

$7 x - 7$

Solution

7(x − 1)

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

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

Solution

3(n² + 7n + 4)

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

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

Solution

6(q² + 5q + 7)

$8 q^{2} + 15 q$

$9 c^{2} + 22 c$

Solution

c(9c + 22)

$13 k^{2} + 5 k$

$17 x^{2} + 7 x$

Solution

x(17x + 7)

$5 c^{2} + 9 c$

$4 q^{2} + 7 q$

Solution

q(4q + 7)

$5 p^{2} + 25 p$

$3 r^{2} + 27 r$

Solution

3r(r + 9)

$24 q^{2} - 12 q$

$30 u^{2} - 10 u$

Solution

10u(3u − 1)

$y z + 4 z$

$a b + 8 b$

Solution

b(a + 8)

$60 x - 6 x^{3}$

$55 y - 11 y^{4}$

Solution

11y(5 − y³)

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

$45 c^{3} - 15 c^{2}$

Solution

15c²(3c − 1)

$4 a^{3} - 4 a b^{2}$

$6 c^{3} - 6 c d^{2}$

Solution

6c(c² − d²)

$30 u^{3} + 80 u^{2}$

$48 x^{3} + 72 x^{2}$

Solution

24x²(2x + 3)

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

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

Solution

18a³(8a³ + 5)

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

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

Solution

10(y² + 5y + 4)

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

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

Solution

12(u² − 3u − 9)

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

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

Solution

5p²(p² − 4p − 3)

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

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

Solution

8c³(c² + 5c − 7)

$−3 n - 24$

$−7 p - 84$

Solution

−7(p + 12)

$−15 a^{2} - 40 a$

$−18 b^{2} - 66 b$

Solution

−6b(3b + 11)

$−10 y^{3} + 60 y^{2}$

$−8 a^{3} + 32 a^{2}$

Solution

−8a²(a − 4)

$−4 u^{5} + 56 u^{3}$

$−9 b^{5} + 63 b^{3}$

Solution

−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 $−5 p^{2} + 150 p .$ 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.

Solution

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

Solution

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.

ⓑ 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.

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

Solution

trinomial

$−6 a^{4}$

$9 x^{3} - 1$

Solution

binomial

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

Determine the Degree of Polynomials

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

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

Solution

$5 m + 9$

$−15$

Solution

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

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$4 p + 11 p$

Solution

15p

$−8 y^{3} - 5 y^{3}$

Add $4 n^{5}, − n^{5}, −6 n^{5}$

Solution

−3n⁵

Subtract $10 x^{2}$ from $3 x^{2}$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

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

Solution

10a² + 4a − 1

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

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

Solution

6y² − 3y + 3

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

Find the sum of $8 q^{3} - 27$ and $q^{2} + 6 q - 2$

Solution

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 $x = 5$

Solution

995

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

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

Solution

2,955

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

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

Solution

−163

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

A pair of glasses is dropped off a bridge $640$ feet above a river. The polynomial $−16 t^{2} + 640$ gives the height of the glasses $t$ seconds after they were dropped. Find the height of the glasses when $t = 6 .$

Solution

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 $x = 20$ mph.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

$6^{3}$

Solution

216

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

${(−0.5)}^{2}$

Solution

0.25

$- 3^{2}$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

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

Solution

p¹³

$2 \cdot 2^{6}$

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

Solution

a⁶

$x \cdot x^{8}$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

${(y^{4})}^{3}$

Solution

y¹²

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

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

Solution

3¹⁰

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

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

${(8 n)}^{2}$

Solution

64n²

${(−5 x)}^{3}$

${(2 a b)}^{8}$

Solution

256a⁸b⁸

${(−10 m n p)}^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(3 a^{5})}^{3}$

Solution

27a¹⁵

${(4 y)}^{2} (8 y)$

${(x^{3})}^{5} {(x^{2})}^{3}$

Solution

x²¹

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

Multiply Monomials

In the following exercises, multiply the monomials.

$(−6 p^{4}) (9 p)$

Solution

−54p⁵

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

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

Solution

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.

$7 (10 - x)$

Solution

70 − 7x

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

$−5 y (125 y^{3} - 1)$

Solution

−625y⁴ + 5y

$(4 n - 5) (2 n^{3})$

Multiply a Binomial by a Binomial

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

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

Solution

a² + 7a + 10

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

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

Solution

6x² − 19x − 7

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

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

Solution

n² + 9n + 8

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

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

Solution

5u² + 37u − 24

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

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

Solution

p² + 11p + 28

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

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

Solution

27c² − 3c − 4

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

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

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

Solution

x³ − 2x² − 24x − 21

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

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

Solution

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.

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

Solution

2⁶ or 64

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

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

Solution

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

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

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

$3^{0}$

Solution

$y^{0}$

${(14 t)}^{0}$

Solution

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

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

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

Solution

$\frac{9}{25}$

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

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

Solution

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

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

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

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

Solution

a²

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

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

Solution

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

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

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

Solution

$\frac{1}{n}$

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

Divide Monomials

In the following exercises, divide the monomials.

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

Solution

9p⁹

$−26 a^{8} \div (2 a^{2})$

$\frac{45 y^{6}}{−15 y^{10}}$

Solution

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

$\frac{−30 x^{8}}{−36 x^{9}}$

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

Solution

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

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

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

Solution

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

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

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

$6^{−2}$

Solution

$\frac{1}{36}$

${(−10)}^{−3}$

$5 \cdot 2^{−4}$

Solution

$\frac{5}{16}$

${(8 n)}^{−1}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$x^{−3} \cdot x^{9}$

Solution

x⁶

$r^{−5} \cdot r^{−4}$

$(u v^{−3}) (u^{−4} v^{−2})$

Solution

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

${(m^{5})}^{−1}$

${(k^{−2})}^{−3}$

Solution

k⁶

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

$\frac{b^{8}}{b^{−2}}$

Solution

b¹⁰

$\frac{n^{−3}}{n^{−5}}$

Convert from Decimal Notation to Scientific Notation

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

$5,300,000$

Solution

5.3 × 10⁶

$0.00814$

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

Solution

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.

$2.9 \times 10^{4}$

Solution

29,000

$1.5 \times 10^{8}$

$3.75 \times 10^{−1}$

Solution

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.

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

Solution

6,000

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

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

Solution

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.

$5 n, 45$

Solution

$8 a, 72$

$12 x^{2}, 20 x^{3}, 36 x^{4}$

Solution

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.

$16 u - 24$

Solution

8(2u − 3)

$15 r + 35$

$6 p^{2} + 6 p$

Solution

6p(p + 1)

$10 c^{2} - 10 c$

$−9 a^{5} - 9 a^{3}$

Solution

−9a³(a² + 1)

$−7 x^{8} - 28 x^{3}$

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

Solution

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?

Solution

ⓐ trinomial
ⓑ 4

In the following exercises, simplify each expression.

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

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

Solution

6x² − 3x + 11

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

$n \cdot n^{4}$

Solution

n⁵

${(10 p^{3} q^{5})}^{2}$

$(8 x y^{3}) (−6 x^{4} y^{6})$

Solution

−48x⁵y⁹

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

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

Solution

s² + 17s + 72

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

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

Solution

55a² − 41a + 6

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

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

Solution

24a² + 34ab − 45b²

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

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

Solution

x¹⁴

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

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

Solution

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

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

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

Solution

3y² − 7x

$4^{−1}$

${(2 y)}^{−3}$

Solution

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

$p^{−3} \cdot p^{−8}$

$\frac{x^{4}}{x^{−5}}$

Solution

x⁹

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

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

$−6 x^{2} - 30 x$

Solution

−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.

Solution

0.000525

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

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

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

Solution

$300,000$

A hiker drops a pebble from a bridge $240$ feet above a canyon. The polynomial $−16 t^{2} + 240$ gives the height of the pebble $t$ seconds a after it was dropped. Find the height when $t = 3 .$

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.