# Chapter 6 Review Exercises - Mathematics

## Chapter 6 Review Exercises

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.

Exercise (PageIndex{1})

1. (11 c^{4}-23 c^{2}+1)
2. (9 p^{3}+6 p^{2}-p-5)
3. (frac{3}{7} x+frac{5}{14})
4. 10
5. 2y−12

Exercise (PageIndex{2})

1. (a^{2}-b^{2})
2. 24(d^{3})
3. (x^{2}+8 x-10)
4. (m^{2} n^{2}-2 m n+6)
5. (7 y^{3}+y^{2}-2 y-4)
1. binomial
2. monomial
3. trinomial
4. trinomial
5. other polynomial

Determine the Degree of Polynomials

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

Exercise (PageIndex{3})

1. (3 x^{2}+9 x+10)
2. 14(a^{2} b c)
3. 6y+1
4. (n^{3}-4 n^{2}+2 n-8)
5. −19

Exercise (PageIndex{4})

1. (5 p^{3}-8 p^{2}+10 p-4)
2. (-20 q^{4})
3. (x^{2}+6 x+12)
4. (23 r^{2} s^{2}-4 r s+5)
5. 100
1. 3
2. 4
3. 2
4. 4
5. 0

In the following exercises, add or subtract the monomials.

Exercise (PageIndex{5})

(5 y^{3}+8 y^{3})

Exercise (PageIndex{6})

(-14 k+19 k)

5k

Exercise (PageIndex{7})

12q−(−6q)

Exercise (PageIndex{8})

−9c−18c

−27c

Exercise (PageIndex{9})

12x−4y−9x

Exercise (PageIndex{2})

(3 m^{2}+7 n^{2}-3 m^{2})

7(n^{2})

Exercise (PageIndex{3})

(6 x^{2} y-4 x+8 x y^{2})

Exercise (PageIndex{4})

13a+b

13a+b

In the following exercises, add or subtract the polynomials.

Exercise (PageIndex{5})

(left(5 x^{2}+12 x+1 ight)+left(6 x^{2}-8 x+3 ight))

Exercise (PageIndex{6})

(left(9 p^{2}-5 p+3 ight)+left(4 p^{2}-4 ight))

(13 p^{2}-5 p-1)

Exercise (PageIndex{7})

(left(10 m^{2}-8 m-1 ight)-left(5 m^{2}+m-2 ight))

Exercise (PageIndex{8})

(left(7 y^{2}-8 y ight)-(y-4))

(7 y^{2}-9 y+4)

Exercise (PageIndex{9})

Subtract
(left(3 s^{2}+10 ight)) from (left(15 s^{2}-2 s+8 ight))

Exercise (PageIndex{10})

Find the sum of (left(a^{2}+6 a+9 ight)) and (left(5 a^{3}-7 ight))

(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.

Exercise (PageIndex{11})

Evaluate (3 y^{2}-y+1) when:

1. y=5
2. y=−1
3. y=0

Exercise (PageIndex{12})

Evaluate 10−12x when:

1. x=3
2. x=0
3. x=−1
1. −26
2. 10
3. 22

Exercise (PageIndex{13})

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.

Exercise (PageIndex{14})

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.

12,000

### Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

Exercise (PageIndex{15})

(10^{4})

Exercise (PageIndex{16})

(17^{1})

17

Exercise (PageIndex{17})

(left(frac{2}{9} ight)^{2})

Exercise (PageIndex{18})

((0.5)^{3})

0.125

Exercise (PageIndex{19})

((-2)^{6})

Exercise (PageIndex{20})

(-2^{6})

−64

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

Exercise (PageIndex{21})

(x^{4} cdot x^{3})

Exercise (PageIndex{22})

(p^{15} cdot p^{16})

(p^{31})

Exercise (PageIndex{23})

(4^{10} cdot 4^{6})

Exercise (PageIndex{24})

8(cdot 8^{5})

(8^{6})

Exercise (PageIndex{25})

(n cdot n^{2} cdot n^{4})

Exercise (PageIndex{26})

(y^{c} cdot y^{3})

(y^{c+3})

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

Exercise (PageIndex{27})

(left(m^{3} ight)^{5})

Exercise (PageIndex{28})

(left(5^{3} ight)^{2})

(5^{6})

Exercise (PageIndex{29})

(left(y^{4} ight)^{x})

Exercise (PageIndex{30})

(left(3^{r} ight)^{s})

(3^{r s})

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

Exercise (PageIndex{31})

((4 a)^{2})

Exercise (PageIndex{32})

((-5 y)^{3})

(-125 y^{3})

Exercise (PageIndex{33})

((2 m n)^{5})

Exercise (PageIndex{34})

((10 x y z)^{3})

1000(x^{3} y^{3} z^{3})

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Exercise (PageIndex{35})

(left(p^{2} ight)^{5} cdotleft(p^{3} ight)^{6})

Exercise (PageIndex{36})

(left(4 a^{3} b^{2} ight)^{3})

64(a^{9} b^{6})

Exercise (PageIndex{37})

((5 x)^{2}(7 x))

Exercise (PageIndex{38})

(left(2 q^{3} ight)^{4}(3 q)^{2})

48(q^{14})

Exercise (PageIndex{39})

(left(frac{1}{3} x^{2} ight)^{2}left(frac{1}{2} x ight)^{3})

Exercise (PageIndex{40})

(left(frac{2}{5} m^{2} n ight)^{3})

(frac{8}{125} m^{6} n^{3})

Multiply Monomials

In the following exercises 8, multiply the monomials.

Exercise (PageIndex{41})

(left(-15 x^{2} ight)left(6 x^{4} ight))

Exercise (PageIndex{42})

(left(-9 n^{7} ight)(-16 n))

144(n^{8})

Exercise (PageIndex{43})

(left(7 p^{5} q^{3} ight)left(8 p q^{9} ight))

Exercise (PageIndex{44})

(left(frac{5}{9} a b^{2} ight)left(27 a b^{3} ight))

15(a^{2} b^{5})

### Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Exercise (PageIndex{45})

7(a+9)

Exercise (PageIndex{46})

−4(y+13)

−4y−52

Exercise (PageIndex{47})

−5(r−2)

Exercise (PageIndex{48})

p(p+3)

(p^{2}+3 p)

Exercise (PageIndex{49})

−m(m+15)

Exercise (PageIndex{50})

−6u(2u+7)

(-12 u^{2}-42 u)

Exercise (PageIndex{51})

9(left(b^{2}+6 b+8 ight))

Exercise (PageIndex{52})

3(q^{2}left(q^{2}-7 q+6 ight) 3)

(3 q^{4}-21 q^{3}+18 q^{2})

Exercise (PageIndex{53})

((5 z-1) z)

Exercise (PageIndex{54})

((b-4) cdot 11)

11b−44

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

1. the Distributive Property,
2. the FOIL method,
3. the Vertical Method.

Exercise (PageIndex{55})

(x−4)(x+10)

Exercise (PageIndex{56})

(6y−7)(2y−5)

1. (12 y^{2}-44y+35)
2. (12 y^{2}-44y+35)
3. (12 y^{2}-44y+35)

In the following exercises, multiply the binomials. Use any method.

Exercise (PageIndex{57})

(x+3)(x+9)

Exercise (PageIndex{58})

(y−4)(y−8)

(y^{2}-12 y+32)

Exercise (PageIndex{59})

(p−7)(p+4)

Exercise (PageIndex{60})

(q+16)(q−3)

(q^{2}+13 q-48)

Exercise (PageIndex{61})

(5m−8)(12m+1)

Exercise (PageIndex{62})

(left(u^{2}+6 ight)left(u^{2}-5 ight))

(u^{4}+u^{2}-30)

Exercise (PageIndex{63})

(9x−y)(6x−5)

Exercise (PageIndex{64})

(8mn+3)(2mn−1)

(16 m^{2} n^{2}-2 m n-3)

Multiply a Trinomial by a Binomial

In the following exercises, multiply using

1. the Distributive Property,
2. the Vertical Method.

Exercise (PageIndex{65})

((n+1)left(n^{2}+5 n-2 ight))

Exercise (PageIndex{66})

((3 x-4)left(6 x^{2}+x-10 ight))

1. (18 x^{3}-21 x^{2}-34 x+40)
2. (18 x^{3}-21 x^{2}-34 x+40)

In the following exercises, multiply. Use either method.

Exercise (PageIndex{67})

((y-2)left(y^{2}-8 y+9 ight))

Exercise (PageIndex{68})

((7 m+1)left(m^{2}-10 m-3 ight))

(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.

Exercise (PageIndex{69})

((c+11)^{2})

Exercise (PageIndex{70})

((q-15)^{2})

(q^{2}-30 q+225)

Exercise (PageIndex{71})

(left(x+frac{1}{3} ight)^{2})

Exercise (PageIndex{72})

((8 u+1)^{2})

(64 u^{2}+16 u+1)

Exercise (PageIndex{73})

(left(3 n^{3}-2 ight)^{2})

Exercise (PageIndex{74})

((4 a-3 b)^{2})

(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.

Exercise (PageIndex{75})

(s−7)(s+7)

Exercise (PageIndex{76})

(left(y+frac{2}{5} ight)left(y-frac{2}{5} ight))

(y^{2}-frac{4}{25})

Exercise (PageIndex{77})

((12 c+13)(12 c-13))

Exercise (PageIndex{78})

(6−r)(6+r)

(36-r^{2})

Exercise (PageIndex{79})

(left(u+frac{3}{4} v ight)left(u-frac{3}{4} v ight))

Exercise (PageIndex{80})

(left(5 p^{4}-4 q^{3} ight)left(5 p^{4}+4 q^{3} ight))

(25 p^{8}-16 q^{6})

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Exercise (PageIndex{81})

((3 m+10)^{2})

Exercise (PageIndex{82})

(6a+11)(6a−11)

(36 a^{2}-121)

Exercise (PageIndex{83})

(5x+y)(x−5y)

Exercise (PageIndex{84})

(left(c^{4}+9 d ight)^{2})

(c^{8}+18 c^{4} d+81 d^{2})

Exercise (PageIndex{85})

(left(p^{5}+q^{5} ight)left(p^{5}-q^{5} ight))

Exercise (PageIndex{86})

(left(a^{2}+4 b ight)left(4 a-b^{2} ight))

(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.

Exercise (PageIndex{87})

(frac{u^{24}}{u^{6}})

Exercise (PageIndex{88})

(frac{10^{25}}{10^{5}})

(10^{20})

Exercise (PageIndex{89})

(frac{3^{4}}{3^{6}})

Exercise (PageIndex{90})

(frac{v^{12}}{v^{48}})

(frac{1}{v^{36}})

Exercise (PageIndex{91})

(frac{x}{x^{5}})

Exercise (PageIndex{92})

(frac{5}{5^{8}})

(frac{1}{5^{7}})

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Exercise (PageIndex{93})

(75^{0})

Exercise (PageIndex{94})

(x^{0})

1

Exercise (PageIndex{95})

(-12^{0})

Exercise (PageIndex{96})

(left(-12^{0} ight)(-12)^{0})

1

Exercise (PageIndex{97})

25(x^{0})

Exercise (PageIndex{98})

((25 x)^{0})

1

Exercise (PageIndex{99})

(19 n^{0}-25 m^{0})

Exercise (PageIndex{100})

((19 n)^{0}-(25 m)^{0})

0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Exercise (PageIndex{101})

(left(frac{2}{5} ight)^{3})

Exercise (PageIndex{102})

(left(frac{m}{3} ight)^{4})

(frac{m^{4}}{81})

Exercise (PageIndex{103})

(left(frac{r}{s} ight)^{8})

Exercise (PageIndex{104})

(left(frac{x}{2 y} ight)^{6})

(frac{x^{6}}{64 y^{6}})

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Exercise (PageIndex{105})

(frac{left(x^{3} ight)^{5}}{x^{9}})

Exercise (PageIndex{106})

(frac{n^{10}}{left(n^{5} ight)^{2}})

1

Exercise (PageIndex{107})

(left(frac{q^{6}}{q^{8}} ight)^{3})

Exercise (PageIndex{108})

(left(frac{r^{8}}{r^{3}} ight)^{4})

(r^{20})

Exercise (PageIndex{109})

(left(frac{c^{2}}{d^{5}} ight)^{9})

Exercise (PageIndex{110})

(left(frac{3 x^{4}}{2 y^{2}} ight)^{5})

(frac{343 x^{20}}{32 y^{10}})

Exercise (PageIndex{111})

(left(frac{v^{3} v^{9}}{v^{6}} ight)^{4})

Exercise (PageIndex{112})

(frac{left(3 n^{2} ight)^{4}left(-5 n^{4} ight)^{3}}{left(-2 n^{5} ight)^{2}})

(-frac{10,125 n^{10}}{4})

Divide Monomials

In the following exercises, divide the monomials.

Exercise (PageIndex{113})

(-65 y^{14} div 5 y^{2})

Exercise (PageIndex{114})

(frac{64 a^{5} b^{9}}{-16 a^{10} b^{3}})

(-frac{4 b^{6}}{a^{5}})

Exercise (PageIndex{115})

(frac{144 x^{15} y^{8} z^{3}}{18 x^{10} y^{2} z^{12}})

Exercise (PageIndex{116})

(frac{left(8 p^{6} q^{2} ight)left(9 p^{3} q^{5} ight)}{16 p^{8} q^{7}})

(frac{9 p}{2})

### Divide Polynomials

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

Exercise (PageIndex{117})

(frac{42 z^{2}-18 z}{6})

Exercise (PageIndex{118})

(left(35 x^{2}-75 x ight) div 5 x)

7x−15

Exercise (PageIndex{119})

(frac{81 n^{4}+105 n^{2}}{-3})

Exercise (PageIndex{120})

(frac{550 p^{6}-300 p^{4}}{10 p^{3}})

(55 p^{3}-30 p)

Exercise (PageIndex{121})

(left(63 x y^{3}+56 x^{2} y^{4} ight) div(7 x y))

Exercise (PageIndex{122})

(frac{96 a^{5} b^{2}-48 a^{4} b^{3}-56 a^{2} b^{4}}{8 a b^{2}})

(12 a^{4}-6 a^{3} b-7 a b^{2})

Exercise (PageIndex{123})

(frac{57 m^{2}-12 m+1}{-3 m})

Exercise (PageIndex{124})

(frac{105 y^{5}+50 y^{3}-5 y}{5 y^{3}})

(21 y^{2}+10-frac{1}{y^{2}})

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

Exercise (PageIndex{125})

(left(k^{2}-2 k-99 ight) div(k+9))

Exercise (PageIndex{126})

(left(v^{2}-16 v+64 ight) div(v-8))

v−8

Exercise (PageIndex{127})

(left(3 x^{2}-8 x-35 ight) div(x-5))

Exercise (PageIndex{128})

(left(n^{2}-3 n-14 ight) div(n+3))

(n-6+frac{4}{n+3})

Exercise (PageIndex{129})

(left(4 m^{3}+m-5 ight) div(m-1))

Exercise (PageIndex{130})

(left(u^{3}-8 ight) div(u-2))

(u^{2}+2 u+4)

### Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Exercise (PageIndex{131})

(9^{-2})

Exercise (PageIndex{132})

((-5)^{-3})

(-frac{1}{125})

Exercise (PageIndex{133})

3(cdot 4^{-3})

Exercise (PageIndex{134})

((6 u)^{-3})

(frac{1}{216 u^{3}})

Exercise (PageIndex{135})

(left(frac{2}{5} ight)^{-1})

Exercise (PageIndex{136})

(left(frac{3}{4} ight)^{-2})

(frac{16}{9})

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Exercise (PageIndex{137})

(p^{-2} cdot p^{8})

Exercise (PageIndex{138})

(q^{-6} cdot q^{-5})

(frac{1}{q^{11}})

Exercise (PageIndex{139})

(left(c^{-2} d ight)left(c^{-3} d^{-2} ight))

Exercise (PageIndex{140})

(left(y^{8} ight)^{-1})

(frac{1}{y^{8}})

Exercise (PageIndex{141})

(left(q^{-4} ight)^{-3})

Exercise (PageIndex{142})

(frac{a^{8}}{a^{12}})

(frac{1}{a^{4}})

Exercise (PageIndex{143})

(frac{n^{5}}{n^{-4}})

Exercise (PageIndex{144})

(frac{r^{-2}}{r^{-3}})

r

Convert from Decimal Notation to Scientific Notation

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

Exercise (PageIndex{145})

8,500,000

Exercise (PageIndex{146})

0.00429

(4.29 imes 10^{-3})

Exercise (PageIndex{147})

The thickness of a dime is about 0.053 inches.

Exercise (PageIndex{148})

In 2015, the population of the world was about 7,200,000,000 people.

(7.2 imes 10^{9})

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

Exercise (PageIndex{149})

(3.8 imes 10^{5})

Exercise (PageIndex{150})

(1.5 imes 10^{10})

15,000,000,000

Exercise (PageIndex{151})

(9.1 imes 10^{-7})

Exercise (PageIndex{152})

(5.5 imes 10^{-1})

0.55

Multiply and Divide Using Scientific Notation

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

Exercise (PageIndex{153})

(left(2 imes 10^{5} ight)left(4 imes 10^{-3} ight))

Exercise (PageIndex{154})

(left(3.5 imes 10^{-2} ight)left(6.2 imes 10^{-1} ight))

0.0217

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

Exercise (PageIndex{155})

(frac{8 imes 10^{5}}{4 imes 10^{-1}})

Exercise (PageIndex{156})

(frac{9 imes 10^{-5}}{3 imes 10^{2}})

0.0000003

## Chapter Practice Test

Exercise (PageIndex{1})

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.

Exercise (PageIndex{2})

(left(12 a^{2}-7 a+4 ight)+left(3 a^{2}+8 a-10 ight))

(15 a^{2}+a-6)

Exercise (PageIndex{3})

(left(9 p^{2}-5 p+1 ight)-left(2 p^{2}-6 ight))

Exercise (PageIndex{4})

(left(-frac{2}{5} ight)^{3})

(-frac{8}{125})

Exercise (PageIndex{5})

(u cdot u^{4})

Exercise (PageIndex{6})

(left(4 a^{3} b^{5} ight)^{2})

16(a^{6} b^{10})

Exercise (PageIndex{7})

(left(-9 r^{4} s^{5} ight)left(4 r s^{7} ight))

Exercise (PageIndex{8})

3(kleft(k^{2}-7 k+13 ight))

(3 k^{3}-21 k^{2}+39 k)

Exercise (PageIndex{9})

((m+6)(m+12))

Exercise (PageIndex{10})

(v−9)(9v−5)

(9 v^{2}-86 v+45)

Exercise (PageIndex{11})

(4c−11)(3c−8)

Exercise (PageIndex{12})

((n-6)left(n^{2}-5 n+4 ight))

(n^{3}-11 n^{2}+34 n-24)

Exercise (PageIndex{13})

((2 x-15 y)(5 x+7 y))

Exercise (PageIndex{14})

((7 p-5)(7 p+5))

(49 p^{2}-25)

Exercise (PageIndex{15})

((9 v-2)^{2})

Exercise (PageIndex{16})

(frac{3^{8}}{3^{10}})

(frac{1}{9})

Exercise (PageIndex{17})

(left(frac{m^{4} cdot m}{m^{3}} ight)^{6})

Exercise (PageIndex{18})

(left(87 x^{15} y^{3} z^{22} ight)^{0})

1

Exercise (PageIndex{19})

(frac{80 c^{8} d^{2}}{16 c d^{10}})

Exercise (PageIndex{20})

(frac{12 x^{2}+42 x-6}{2 x})

(6 x+21-frac{3}{x})

Exercise (PageIndex{21})

(left(70 x y^{4}+95 x^{3} y ight) div 5 x y)

Exercise (PageIndex{22})

(frac{64 x^{3}-1}{4 x-1})

(16 x^{2}+4 x+1)

Exercise (PageIndex{23})

(left(y^{2}-5 y-18 ight) div(y+3))

Exercise (PageIndex{24})

(5^{-2})

(frac{1}{25})

Exercise (PageIndex{25})

((4 m)^{-3})

Exercise (PageIndex{26})

(q^{-4} cdot q^{-5})

(frac{1}{q^{9}})

Exercise (PageIndex{27})

(frac{n^{-2}}{n^{-10}})

Exercise (PageIndex{28})

Convert 83,000,000 to scientific notation.

(8.3 imes 10^{7})

Exercise (PageIndex{29})

Convert (6.91 imes 10^{-5}) to decimal form.

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

Exercise (PageIndex{30})

(left(3.4 imes 10^{9} ight)left(2.2 imes 10^{-5} ight))

74,800

Exercise (PageIndex{31})

(frac{8.4 imes 10^{-3}}{4 imes 10^{3}})

Exercise (PageIndex{32})

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.