Chapter 6 Review Exercises
- Page ID
- 30257
Chapter 6 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.
- \(11 c^{4}-23 c^{2}+1\)
- \(9 p^{3}+6 p^{2}-p-5\)
- \(\frac{3}{7} x+\frac{5}{14}\)
- 10
- 2y−12
- \(a^{2}-b^{2}\)
- 24\(d^{3}\)
- \(x^{2}+8 x-10\)
- \(m^{2} n^{2}-2 m n+6\)
- \(7 y^{3}+y^{2}-2 y-4\)
- Answer
-
- binomial
- monomial
- trinomial
- trinomial
- other polynomial
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
- \(3 x^{2}+9 x+10\)
- 14\(a^{2} b c\)
- 6y+1
- \(n^{3}-4 n^{2}+2 n-8\)
- −19
- \(5 p^{3}-8 p^{2}+10 p-4\)
- \(-20 q^{4}\)
- \(x^{2}+6 x+12\)
- \(23 r^{2} s^{2}-4 r s+5\)
- 100
- Answer
-
- 3
- 4
- 2
- 4
- 0
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
\(5 y^{3}+8 y^{3}\)
\(-14 k+19 k\)
- Answer
-
5k
12q−(−6q)
−9c−18c
- Answer
-
−27c
12x−4y−9x
\(3 m^{2}+7 n^{2}-3 m^{2}\)
- Answer
-
7\(n^{2}\)
\(6 x^{2} y-4 x+8 x y^{2}\)
13a+b
- Answer
-
13a+b
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
\(\left(5 x^{2}+12 x+1\right)+\left(6 x^{2}-8 x+3\right)\)
\(\left(9 p^{2}-5 p+3\right)+\left(4 p^{2}-4\right)\)
- Answer
-
\(13 p^{2}-5 p-1\)
\(\left(10 m^{2}-8 m-1\right)-\left(5 m^{2}+m-2\right)\)
\(\left(7 y^{2}-8 y\right)-(y-4)\)
- Answer
-
\(7 y^{2}-9 y+4\)
Subtract
\(\left(3 s^{2}+10\right)\) from \(\left(15 s^{2}-2 s+8\right)\)
Find the sum of \(\left(a^{2}+6 a+9\right)\) and \(\left(5 a^{3}-7\right)\)
- Answer
-
\(5 a^{3}+a^{2}+6 a+2\)
Evaluate a Polynomial for a Given Value of the Variable
In the following exercises, evaluate each polynomial for the given value.
Evaluate \(3 y^{2}-y+1\) when:
- y=5
- y=−1
- y=0
Evaluate 10−12x when:
- x=3
- x=0
- x=−1
- Answer
-
- −26
- 10
- 22
Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial \(-16 t^{2}+200\) gives the height of a stone t seconds after it is dropped from the cliff. Find the height after t=3 seconds.
A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial \(-4 p^{2}+460 p\). Find the revenue received when p=75 dollars.
- Answer
-
12,000
Use Multiplication Properties of Exponents
Simplify Expressions with Exponents
In the following exercises, simplify.
\(10^{4}\)
\(17^{1}\)
- Answer
-
17
\(\left(\frac{2}{9}\right)^{2}\)
\((0.5)^{3}\)
- Answer
-
0.125
\((-2)^{6}\)
\(-2^{6}\)
- Answer
-
−64
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression.
\(x^{4} \cdot x^{3}\)
\(p^{15} \cdot p^{16}\)
- Answer
-
\(p^{31}\)
\(4^{10} \cdot 4^{6}\)
8\(\cdot 8^{5}\)
- Answer
-
\(8^{6}\)
\(n \cdot n^{2} \cdot n^{4}\)
\(y^{c} \cdot y^{3}\)
- Answer
-
\(y^{c+3}\)
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression.
\(\left(m^{3}\right)^{5}\)
\(\left(5^{3}\right)^{2}\)
- Answer
-
\(5^{6}\)
\(\left(y^{4}\right)^{x}\)
\(\left(3^{r}\right)^{s}\)
- Answer
-
\(3^{r s}\)
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
\((4 a)^{2}\)
\((-5 y)^{3}\)
- Answer
-
\(-125 y^{3}\)
\((2 m n)^{5}\)
\((10 x y z)^{3}\)
- Answer
-
1000\(x^{3} y^{3} z^{3}\)
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
\(\left(p^{2}\right)^{5} \cdot\left(p^{3}\right)^{6}\)
\(\left(4 a^{3} b^{2}\right)^{3}\)
- Answer
-
64\(a^{9} b^{6}\)
\((5 x)^{2}(7 x)\)
\(\left(2 q^{3}\right)^{4}(3 q)^{2}\)
- Answer
-
48\(q^{14}\)
\(\left(\frac{1}{3} x^{2}\right)^{2}\left(\frac{1}{2} x\right)^{3}\)
\(\left(\frac{2}{5} m^{2} n\right)^{3}\)
- Answer
-
\(\frac{8}{125} m^{6} n^{3}\)
Multiply Monomials
In the following exercises 8, multiply the monomials.
\(\left(-15 x^{2}\right)\left(6 x^{4}\right)\)
\(\left(-9 n^{7}\right)(-16 n)\)
- Answer
-
144\(n^{8}\)
\(\left(7 p^{5} q^{3}\right)\left(8 p q^{9}\right)\)
\(\left(\frac{5}{9} a b^{2}\right)\left(27 a b^{3}\right)\)
- Answer
-
15\(a^{2} b^{5}\)
Multiply Polynomials
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
7(a+9)
−4(y+13)
- Answer
-
−4y−52
−5(r−2)
p(p+3)
- Answer
-
\(p^{2}+3 p\)
−m(m+15)
−6u(2u+7)
- Answer
-
\(-12 u^{2}-42 u\)
9\(\left(b^{2}+6 b+8\right)\)
3\(q^{2}\left(q^{2}-7 q+6\right) 3\)
- Answer
-
\(3 q^{4}-21 q^{3}+18 q^{2}\)
\((5 z-1) z\)
\((b-4) \cdot 11\)
- Answer
-
11b−44
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using:
- the Distributive Property,
- the FOIL method,
- the Vertical Method.
(x−4)(x+10)
(6y−7)(2y−5)
- Answer
-
- \(12 y^{2}-44y+35\)
- \(12 y^{2}-44y+35\)
- \(12 y^{2}-44y+35\)
In the following exercises, multiply the binomials. Use any method.
(x+3)(x+9)
(y−4)(y−8)
- Answer
-
\(y^{2}-12 y+32\)
(p−7)(p+4)
(q+16)(q−3)
- Answer
-
\(q^{2}+13 q-48\)
(5m−8)(12m+1)
\(\left(u^{2}+6\right)\left(u^{2}-5\right)\)
- Answer
-
\(u^{4}+u^{2}-30\)
(9x−y)(6x−5)
(8mn+3)(2mn−1)
- Answer
-
\(16 m^{2} n^{2}-2 m n-3\)
Multiply a Trinomial by a Binomial
In the following exercises, multiply using
- the Distributive Property,
- the Vertical Method.
\((n+1)\left(n^{2}+5 n-2\right)\)
\((3 x-4)\left(6 x^{2}+x-10\right)\)
- Answer
-
- \(18 x^{3}-21 x^{2}-34 x+40\)
- \(18 x^{3}-21 x^{2}-34 x+40\)
In the following exercises, multiply. Use either method.
\((y-2)\left(y^{2}-8 y+9\right)\)
\((7 m+1)\left(m^{2}-10 m-3\right)\)
- Answer
-
\(7 m^{3}-69 m^{2}-31 m-3\)
Special Products
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
\((c+11)^{2}\)
\((q-15)^{2}\)
- Answer
-
\(q^{2}-30 q+225\)
\(\left(x+\frac{1}{3}\right)^{2}\)
\((8 u+1)^{2}\)
- Answer
-
\(64 u^{2}+16 u+1\)
\(\left(3 n^{3}-2\right)^{2}\)
\((4 a-3 b)^{2}\)
- Answer
-
\(16 a^{2}-24 a b+9 b^{2}\)
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
(s−7)(s+7)
\(\left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right)\)
- Answer
-
\(y^{2}-\frac{4}{25}\)
\((12 c+13)(12 c-13)\)
(6−r)(6+r)
- Answer
-
\(36-r^{2}\)
\(\left(u+\frac{3}{4} v\right)\left(u-\frac{3}{4} v\right)\)
\(\left(5 p^{4}-4 q^{3}\right)\left(5 p^{4}+4 q^{3}\right)\)
- Answer
-
\(25 p^{8}-16 q^{6}\)
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
\((3 m+10)^{2}\)
(6a+11)(6a−11)
- Answer
-
\(36 a^{2}-121\)
(5x+y)(x−5y)
\(\left(c^{4}+9 d\right)^{2}\)
- Answer
-
\(c^{8}+18 c^{4} d+81 d^{2}\)
\(\left(p^{5}+q^{5}\right)\left(p^{5}-q^{5}\right)\)
\(\left(a^{2}+4 b\right)\left(4 a-b^{2}\right)\)
- Answer
-
\(4 a^{3}+3 a^{2} b-4 b^{3}\)
Divide Monomials
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
\(\frac{u^{24}}{u^{6}}\)
\(\frac{10^{25}}{10^{5}}\)
- Answer
-
\(10^{20}\)
\(\frac{3^{4}}{3^{6}}\)
\(\frac{v^{12}}{v^{48}}\)
- Answer
-
\(\frac{1}{v^{36}}\)
\(\frac{x}{x^{5}}\)
\(\frac{5}{5^{8}}\)
- Answer
-
\(\frac{1}{5^{7}}\)
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
\(75^{0}\)
\(x^{0}\)
- Answer
-
1
\(-12^{0}\)
\(\left(-12^{0}\right)(-12)^{0}\)
- Answer
-
1
25\(x^{0}\)
\((25 x)^{0}\)
- Answer
-
1
\(19 n^{0}-25 m^{0}\)
\((19 n)^{0}-(25 m)^{0}\)
- Answer
-
0
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
\(\left(\frac{2}{5}\right)^{3}\)
\(\left(\frac{m}{3}\right)^{4}\)
- Answer
-
\(\frac{m^{4}}{81}\)
\(\left(\frac{r}{s}\right)^{8}\)
\(\left(\frac{x}{2 y}\right)^{6}\)
- Answer
-
\(\frac{x^{6}}{64 y^{6}}\)
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
\(\frac{\left(x^{3}\right)^{5}}{x^{9}}\)
\(\frac{n^{10}}{\left(n^{5}\right)^{2}}\)
- Answer
-
1
\(\left(\frac{q^{6}}{q^{8}}\right)^{3}\)
\(\left(\frac{r^{8}}{r^{3}}\right)^{4}\)
- Answer
-
\(r^{20}\)
\(\left(\frac{c^{2}}{d^{5}}\right)^{9}\)
\(\left(\frac{3 x^{4}}{2 y^{2}}\right)^{5}\)
- Answer
-
\(\frac{343 x^{20}}{32 y^{10}}\)
\(\left(\frac{v^{3} v^{9}}{v^{6}}\right)^{4}\)
\(\frac{\left(3 n^{2}\right)^{4}\left(-5 n^{4}\right)^{3}}{\left(-2 n^{5}\right)^{2}}\)
- Answer
-
\(-\frac{10,125 n^{10}}{4}\)
Divide Monomials
In the following exercises, divide the monomials.
\(-65 y^{14} \div 5 y^{2}\)
\(\frac{64 a^{5} b^{9}}{-16 a^{10} b^{3}}\)
- Answer
-
\(-\frac{4 b^{6}}{a^{5}}\)
\(\frac{144 x^{15} y^{8} z^{3}}{18 x^{10} y^{2} z^{12}}\)
\(\frac{\left(8 p^{6} q^{2}\right)\left(9 p^{3} q^{5}\right)}{16 p^{8} q^{7}}\)
- Answer
-
\(\frac{9 p}{2}\)
Divide Polynomials
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
\(\frac{42 z^{2}-18 z}{6}\)
\(\left(35 x^{2}-75 x\right) \div 5 x\)
- Answer
-
7x−15
\(\frac{81 n^{4}+105 n^{2}}{-3}\)
\(\frac{550 p^{6}-300 p^{4}}{10 p^{3}}\)
- Answer
-
\(55 p^{3}-30 p\)
\(\left(63 x y^{3}+56 x^{2} y^{4}\right) \div(7 x y)\)
\(\frac{96 a^{5} b^{2}-48 a^{4} b^{3}-56 a^{2} b^{4}}{8 a b^{2}}\)
- Answer
-
\(12 a^{4}-6 a^{3} b-7 a b^{2}\)
\(\frac{57 m^{2}-12 m+1}{-3 m}\)
\(\frac{105 y^{5}+50 y^{3}-5 y}{5 y^{3}}\)
- Answer
-
\(21 y^{2}+10-\frac{1}{y^{2}}\)
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.
\(\left(k^{2}-2 k-99\right) \div(k+9)\)
\(\left(v^{2}-16 v+64\right) \div(v-8)\)
- Answer
-
v−8
\(\left(3 x^{2}-8 x-35\right) \div(x-5)\)
\(\left(n^{2}-3 n-14\right) \div(n+3)\)
- Answer
-
\(n-6+\frac{4}{n+3}\)
\(\left(4 m^{3}+m-5\right) \div(m-1)\)
\(\left(u^{3}-8\right) \div(u-2)\)
- Answer
-
\(u^{2}+2 u+4\)
Integer Exponents and Scientific Notation
Use the Definition of a Negative Exponent
In the following exercises, simplify.
\(9^{-2}\)
\((-5)^{-3}\)
- Answer
-
\(-\frac{1}{125}\)
3\(\cdot 4^{-3}\)
\((6 u)^{-3}\)
- Answer
-
\(\frac{1}{216 u^{3}}\)
\(\left(\frac{2}{5}\right)^{-1}\)
\(\left(\frac{3}{4}\right)^{-2}\)
- Answer
-
\(\frac{16}{9}\)
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
\(p^{-2} \cdot p^{8}\)
\(q^{-6} \cdot q^{-5}\)
- Answer
-
\(\frac{1}{q^{11}}\)
\(\left(c^{-2} d\right)\left(c^{-3} d^{-2}\right)\)
\(\left(y^{8}\right)^{-1}\)
- Answer
-
\(\frac{1}{y^{8}}\)
\(\left(q^{-4}\right)^{-3}\)
\(\frac{a^{8}}{a^{12}}\)
- Answer
-
\(\frac{1}{a^{4}}\)
\(\frac{n^{5}}{n^{-4}}\)
\(\frac{r^{-2}}{r^{-3}}\)
- Answer
-
r
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
8,500,000
0.00429
- Answer
-
\(4.29 \times 10^{-3}\)
The thickness of a dime is about 0.053 inches.
In 2015, the population of the world was about 7,200,000,000 people.
- Answer
-
\(7.2 \times 10^{9}\)
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
\(3.8 \times 10^{5}\)
\(1.5 \times 10^{10}\)
- Answer
-
15,000,000,000
\(9.1 \times 10^{-7}\)
\(5.5 \times 10^{-1}\)
- Answer
-
0.55
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.
\(\left(2 \times 10^{5}\right)\left(4 \times 10^{-3}\right)\)
\(\left(3.5 \times 10^{-2}\right)\left(6.2 \times 10^{-1}\right)\)
- Answer
-
0.0217
In the following exercises, divide and write your answer in decimal form.
\(\frac{8 \times 10^{5}}{4 \times 10^{-1}}\)
\(\frac{9 \times 10^{-5}}{3 \times 10^{2}}\)
- Answer
-
0.0000003
Chapter Practice Test
For the polynomial \(10 x^{4}+9 y^{2}-1\)
ⓐ Is it a monomial, binomial, or trinomial?
ⓑ What is its degree?
In the following exercises, simplify each expression.
\(\left(12 a^{2}-7 a+4\right)+\left(3 a^{2}+8 a-10\right)\)
- Answer
-
\(15 a^{2}+a-6\)
\(\left(9 p^{2}-5 p+1\right)-\left(2 p^{2}-6\right)\)
\(\left(-\frac{2}{5}\right)^{3}\)
- Answer
-
\(-\frac{8}{125}\)
\(u \cdot u^{4}\)
\(\left(4 a^{3} b^{5}\right)^{2}\)
- Answer
-
16\(a^{6} b^{10}\)
\(\left(-9 r^{4} s^{5}\right)\left(4 r s^{7}\right)\)
3\(k\left(k^{2}-7 k+13\right)\)
- Answer
-
\(3 k^{3}-21 k^{2}+39 k\)
\((m+6)(m+12)\)
(v−9)(9v−5)
- Answer
-
\(9 v^{2}-86 v+45\)
(4c−11)(3c−8)
\((n-6)\left(n^{2}-5 n+4\right)\)
- Answer
-
\(n^{3}-11 n^{2}+34 n-24\)
\((2 x-15 y)(5 x+7 y)\)
\((7 p-5)(7 p+5)\)
- Answer
-
\(49 p^{2}-25\)
\((9 v-2)^{2}\)
\(\frac{3^{8}}{3^{10}}\)
- Answer
-
\(\frac{1}{9}\)
\(\left(\frac{m^{4} \cdot m}{m^{3}}\right)^{6}\)
\(\left(87 x^{15} y^{3} z^{22}\right)^{0}\)
- Answer
-
1
\(\frac{80 c^{8} d^{2}}{16 c d^{10}}\)
\(\frac{12 x^{2}+42 x-6}{2 x}\)
- Answer
-
\(6 x+21-\frac{3}{x}\)
\(\left(70 x y^{4}+95 x^{3} y\right) \div 5 x y\)
\(\frac{64 x^{3}-1}{4 x-1}\)
- Answer
-
\(16 x^{2}+4 x+1\)
\(\left(y^{2}-5 y-18\right) \div(y+3)\)
\(5^{-2}\)
- Answer
-
\(\frac{1}{25}\)
\((4 m)^{-3}\)
\(q^{-4} \cdot q^{-5}\)
- Answer
-
\(\frac{1}{q^{9}}\)
\(\frac{n^{-2}}{n^{-10}}\)
Convert 83,000,000 to scientific notation.
- Answer
-
\(8.3 \times 10^{7}\)
Convert \(6.91 \times 10^{-5}\) to decimal form.
In the following exercises, simplify, and write your answer in decimal form.
\(\left(3.4 \times 10^{9}\right)\left(2.2 \times 10^{-5}\right)\)
- Answer
-
74,800
\(\frac{8.4 \times 10^{-3}}{4 \times 10^{3}}\)
A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial \(-16 t^{2}+1000\) gives the height of the package t seconds a after it was dropped. Find the height when t=6 seconds.
- Answer
-
424 feet