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10.E: Polynomials (Exercises)

Last updated

May 2, 2022
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- 10.9: Integer Exponents and Scientific Notation (Part 2)
- 10.S: Polynomials (Summary)

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10.1 - 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.

y²+ 8y − 20
−6a⁴
9x³ − 1
n³ − 3n² + 3n − 1

Determine the Degree of Polynomials

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

16x² − 40x − 25
5m + 9
−15
y² + 6y³ + 9y⁴

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

4p + 11p
−8y³− 5y³
Add 4n⁵, −n⁵, −6n⁵
Subtract 10x² from 3x²

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

(4a 2 + 9a − 11) + (6a 2 − 5a + 10)
(8m 2 + 12m − 5) − (2m 2 − 7m − 1)
(y 2 − 3y + 12) + (5y 2 − 9)
(5u 2 + 8u) − (4u − 7)
Find the sum of 8q³ − 27 and q² + 6q − 2
Find the difference of x²+ 6x + 8 and x² − 8x + 15

Evaluate a Polynomial for a Given Value of the Variable

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

200x − $\dfrac{1}{5} x^{2}$ when x = 5
200x − $\dfrac{1}{5} x^{2}$ when x = 0
200x − $\dfrac{1}{5} x^{2}$ when x = 15
5 + 40x − $\dfrac{1}{2} x^{2}$ when x = 10
5 + 40x − $\dfrac{1}{2} x^{2}$ when x = −4
5 + 40x − $\dfrac{1}{2} x^{2}$ when x = 0
A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial −16t² + 640 gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when t = 6.
The fuel efficiency (in miles per gallon) of a bus going at a speed of x miles per hour is given by the polynomial $− \dfrac{1}{160} x^{2} + \dfrac{1}{2} x$ . Find the fuel efficiency when x = 20 mph.

10.2 - Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

6³
$\left(\dfrac{1}{2}\right)^{4}$
(−0.5)²
−3²

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

p³ • p¹⁰
2 • 2⁶
a • a² • a³
x • x⁸

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

(y⁴)³
(r³)²
(3²)⁵
(a¹⁰)^y

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

(8n)²
(−5x)³
(2ab)⁸
(−10mnp)⁴

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

(3a⁵)³
(4y)²(8y)
(x³)⁵(x²)³
(5st²)³(2s³t⁴)²

Multiply Monomials

In the following exercises, multiply the monomials.

(−6p⁴)(9p)
$\left(\dfrac{1}{3} c^{2}\right)$ (30c⁸)
(8x²y⁵)(7xy⁶)
$\left(\dfrac{2}{3} m^{3} n^{6}\right) \left(\dfrac{1}{6} m^{4} n^{4}\right)$

10.3 - Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

7(10 − x)
a²(a² − 9a − 36)
−5y(125y³ − 1)
(4n − 5)(2n³)

Multiply a Binomial by a Binomial

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

(a + 5)(a + 2)
(y − 4)(y + 12)
(3x + 1)(2x − 7)
(6p − 11)(3p − 10)
(n + 8)(n + 1)
(k + 6)(k − 9)
(5u − 3)(u + 8)
(2y − 9)(5y − 7)
(p + 4)(p + 7)
(x − 8)(x + 9)
(3c + 1)(9c − 4)
(10a − 1)(3a − 3)

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

(x + 1)(x²− 3x − 21)
(5b − 2)(3b² + b − 9)
(m + 6)(m² − 7m − 30)
(4y − 1)(6y² − 12y + 5)

10.4 - Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

$\dfrac{2^{8}}{2^{2}}$
$\dfrac{a^{6}}{a}$
$\dfrac{n^{3}}{n^{12}}$
$\dfrac{x}{x^{5}}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

3⁰
y⁰
(14t)⁰
12a⁰ − 15b⁰

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

$\left(\dfrac{3}{5}\right)^{2}$
$\left(\dfrac{x}{2}\right)^{5}$
$\left(\dfrac{5m}{n}\right)^{3}$
$\left(\dfrac{s}{10t}\right)^{2}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\dfrac{(a^{3})^{2}}{a^{4}}$
$\dfrac{u^{3}}{u^{2} \cdot u^{4}}$
$\left(\dfrac{x}{x^{9}}\right)^{5}$
$\left(\dfrac{p^{4} \cdot p^{5}}{p^{3}}\right)^{2}$
$\dfrac{(n^{5})^{3}}{(n^{2})^{8}}$
$\left(\dfrac{5s^{2}}{4t}\right)^{3}$

Divide Monomials

In the following exercises, divide the monomials.

72p¹²÷ 8p³
−26a⁸ ÷ (2a²)
$\dfrac{45y^{6}}{−15y^{10}}$
$\dfrac{−30x^{8}}{−36x^{9}}$
$\dfrac{28a^{9} b}{7a^{4} b^{3}}$
$\dfrac{11u^{6} v^{3}}{55u^{2} v^{8}}$
$\dfrac{(5m^{9} n^{3})(8m^{3} n^{2})}{(10mn^{4})(m^{2} n^{5})}$
$\dfrac{42r^{2} s^{4}}{6rs^{3}} − \dfrac{54rs^{2}}{9s}$

10.5 - Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

6⁻²
(−10)⁻³
5 • 2⁻⁴
(8n)⁻¹

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

x⁻³ • x⁹
r⁻⁵•r⁻⁴
(uv⁻³)(u⁻⁴v⁻²)
(m⁵)⁻¹
(k⁻²)⁻³
$\dfrac{q^{4}}{q^{20}}$
$\dfrac{b^{8}}{b^{−2}}$
$\dfrac{n^{−3}}{n^{−5}}$

Convert from Decimal Notation to Scientific Notation

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

5,300,000
0.00814
The thickness of a piece of paper is about 0.097 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 × 10⁴
1.5 × 10⁸
3.75 × 10⁻¹
9.413 × 10⁻⁵

Multiply and Divide Using Scientific Notation

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

(3 × 10⁷)(2 × 10⁻⁴)
(1.5 × 10⁻³)(4.8 × 10⁻¹)
$\dfrac{6 \times 10^{9}}{2 \times 10^{−1}}$
$\dfrac{9 \times 10^{-3}}{1 \times 10^{−6}}$

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.

5n, 45
8a, 72
12x², 20x³, 36x⁴
9y⁴, 21y⁵, 15y⁶

Factor the Greatest Common Factor from a Polynomial

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

16u − 24
15r + 35
6p² + 6p
10c² − 10c
−9a⁵ − 9a³
−7x⁸ − 28x³
5y² − 55y + 45
2q⁵ − 16q³ + 30q²

PRACTICE TEST

For the polynomial 8y⁴ − 3y² + 1
1. Is it a monomial, binomial, or trinomial?
2. What is its degree?

In the following exercises, simplify each expression.

(5a² + 2a − 12) + (9a² + 8a − 4)
(10x²− 3x + 5) − (4x² − 6)
$\left(− \dfrac{3}{4}\right)^{3}$
n • n⁴
(10p³q⁵)²
(8xy³)(−6x⁴y⁶)
4u(u² − 9u + 1)
(s + 8)(s + 9)
(m + 3)(7m − 2)
(11a − 6)(5a − 1)
(n − 8)(n 2 − 4n + 11)
(4a + 9b)(6a − 5b)
$\dfrac{5^{6}}{5^{8}}$
$\left(\dfrac{x^{3} \cdot x^{9}}{x^{5}}\right)^{2}$
(47a¹⁸b²³c⁵)⁰
$\dfrac{24r^{3}s}{6r^{2} s^{7}}$
$\dfrac{8y^{2} − 16y + 20}{4y}$
(15xy³ − 35x²y) ÷ 5xy
4⁻¹
(2y)⁻³
p⁻³ • p⁻⁸
$\dfrac{x^{4}}{x^{−5}}$
(2.4 × 10⁸)(2 × 10⁻⁵)

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

80a³ + 120a²+ 40a
−6x² − 30x
Convert 5.25 × 10⁻⁴ to decimal form.

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

$\dfrac{9 \times 10^{4}}{3 \times 10^{−1}}$
A hiker drops a pebble from a bridge 240 feet above a canyon. The polynomial −16t² + 240 gives the height of the pebble t seconds a after it was dropped. Find the height when t = 3.
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.

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

10.1 - Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials

Determine the Degree of Polynomials

Add and Subtract Monomials

Add and Subtract Polynomials

Evaluate a Polynomial for a Given Value of the Variable

10.2 - Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

Simplify Expressions Using the Product Property of Exponents

Simplify Expressions Using the Power Property of Exponents

Simplify Expressions Using the Product to a Power Property

Simplify Expressions by Applying Several Properties

Multiply Monomials

10.3 - Multiply Polynomials

Multiply a Polynomial by a Monomial

Multiply a Binomial by a Binomial

Multiply a Trinomial by a Binomial

10.4 - Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

Simplify Expressions with Zero Exponents

Simplify Expressions Using the Quotient to a Power Property

Simplify Expressions by Applying Several Properties

Divide Monomials

10.5 - Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

Simplify Expressions with Integer Exponents

Convert from Decimal Notation to Scientific Notation

Convert Scientific Notation to Decimal Form

Multiply and Divide Using Scientific Notation

10.6 - Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions

Factor the Greatest Common Factor from a Polynomial

PRACTICE TEST

Contributors and Attributions

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