10.E: Polynomials (Exercises)
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.
- y 2 + 8y − 20
- −6a 4
- 9x 3 − 1
- n 3 − 3n 2 + 3n − 1
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
- 16x 2 − 40x − 25
- 5m + 9
- −15
- y 2 + 6y 3 + 9y 4
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
- 4p + 11p
- −8y 3 − 5y 3
- Add 4n 5 , −n 5 , −6n 5
- Subtract 10x 2 from 3x 2
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 8q 3 − 27 and q 2 + 6q − 2
- Find the difference of x 2 + 6x + 8 and x 2 − 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 −16t 2 + 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.
- 6 3
- \(\left(\dfrac{1}{2}\right)^{4}\)
- (−0.5) 2
- −3 2
Simplify Expressions Using the Product Property of Exponents
In the following exercises, simplify each expression.
- p 3 • p 10
- 2 • 2 6
- a • a 2 • a 3
- x • x 8
Simplify Expressions Using the Power Property of Exponents
In the following exercises, simplify each expression.
- (y 4 ) 3
- (r 3 ) 2
- (3 2 ) 5
- (a 10 ) 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.
- (3a 5 ) 3
- (4y) 2 (8y)
- (x 3 ) 5 (x 2 ) 3
- (5st 2 ) 3 (2s 3 t 4 ) 2
Multiply Monomials
In the following exercises, multiply the monomials.
- (−6p 4 )(9p)
- \(\left(\dfrac{1}{3} c^{2}\right)\)(30c 8 )
- (8x 2 y 5 )(7xy 6 )
- \(\left(\dfrac{2}{3} m^{3} n^{6}\right) \left(\dfrac{1}{6} m^{4} n^{4}\right)\)
10.3 - Multiply Polynomials
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
- 7(10 − x)
- a 2 (a 2 − 9a − 36)
- −5y(125y 3 − 1)
- (4n − 5)(2n 3 )
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)(x 2 − 3x − 21)
- (5b − 2)(3b 2 + b − 9)
- (m + 6)(m 2 − 7m − 30)
- (4y − 1)(6y 2 − 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.
- 3 0
- y 0
- (14t) 0
- 12a 0 − 15b 0
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.
- 72p 12 ÷ 8p 3
- −26a 8 ÷ (2a 2 )
- \(\dfrac{45y^{6}}{−15y^{10}}\)
- \(\dfrac{−30x^{8}}{−36x^{9}}\)
- \(\dfrac{28a^{9} b}{7a^{4} b^{3}}\)
- \(\dfrac{11u^{6} v^{3}}{55u^{2} v^{8}}\)
- \(\dfrac{(5m^{9} n^{3})(8m^{3} n^{2})}{(10mn^{4})(m^{2} n^{5})}\)
- \(\dfrac{42r^{2} s^{4}}{6rs^{3}} − \dfrac{54rs^{2}}{9s}\)
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 • x 9
- r −5 •r −4
- (uv −3 )(u −4 v −2 )
- (m 5 ) −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 × 10 4
- 1.5 × 10 8
- 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 × 10 7 )(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
- 12x 2 , 20x 3 , 36x 4
- 9y 4 , 21y 5 , 15y 6
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
- 16u − 24
- 15r + 35
- 6p 2 + 6p
- 10c 2 − 10c
- −9a 5 − 9a 3
- −7x 8 − 28x 3
- 5y 2 − 55y + 45
- 2q 5 − 16q 3 + 30q 2
PRACTICE TEST
-
For the polynomial 8y
4
− 3y
2
+ 1
- Is it a monomial, binomial, or trinomial?
- What is its degree?
In the following exercises, simplify each expression.
- (5a 2 + 2a − 12) + (9a 2 + 8a − 4)
- (10x 2 − 3x + 5) − (4x 2 − 6)
- \(\left(− \dfrac{3}{4}\right)^{3}\)
- n • n 4
- (10p 3 q 5 ) 2
- (8xy 3 )(−6x 4 y 6 )
- 4u(u 2 − 9u + 1)
- (s + 8)(s + 9)
- (m + 3)(7m − 2)
- (11a − 6)(5a − 1)
- (n − 8)(n 2 − 4n + 11)
- (4a + 9b)(6a − 5b)
- \(\dfrac{5^{6}}{5^{8}}\)
- \(\left(\dfrac{x^{3} \cdot x^{9}}{x^{5}}\right)^{2}\)
- (47a 18 b 23 c 5 ) 0
- \(\dfrac{24r^{3}s}{6r^{2} s^{7}}\)
- \(\dfrac{8y^{2} − 16y + 20}{4y}\)
- (15xy 3 − 35x 2 y) ÷ 5xy
- 4 −1
- (2y) −3
- p −3 • p −8
- \(\dfrac{x^{4}}{x^{−5}}\)
- (2.4 × 10 8 )(2 × 10 −5 )
In the following exercises, factor the greatest common factor from each polynomial.
- 80a 3 + 120a 2 + 40a
- −6x 2 − 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 −16t 2 + 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 ."