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.
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)
- \(\dfrac{6 \times 10^{9}}{2 \times 10^{−1}}\)
- \(\dfrac{9 \times 10^{-3}}{1 \times 10^{−6}}\)
PRACTICE TEST
- For the polynomial 8y4 − 3y2 + 1
- Is it a monomial, binomial, or trinomial?
- 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.