Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

Chapter 6

Rational Expressions, Equations, and Functions - all with Video Answers

Educators

AG

Section 1

Rational Expressions and Functions: Multiplying and Dividing

Problem 1

Choose from the following list the word that best completes each statement. Words may be used more than once or not at all.
domain,factor,invert,range,rational,reciprocal
The expression $\frac{x-4}{5 x}$ is an example of a(n)_____ expression.

AG

Ankit Gupta

Numerade Educator

Problem 2

Choose from the following list the word that best completes each statement. Words may be used more than once or not at all.
domain,factor,invert,range,rational,reciprocal
Any value that makes the denominator of a rational function 0 is not in the _____ of that function.

AG

Ankit Gupta

Numerade Educator

Problem 3

Choose from the following list the word that best completes each statement. Words may be used more than once or not at all.
domain,factor,invert,range,rational,reciprocal
When simplifying rational expressions, remove $\mathrm{a}(\mathrm{n})$_____equal to 1

AG

Ankit Gupta

Numerade Educator

Problem 4

Choose from the following list the word that best completes each statement. Words may be used more than once or not at all.
domain,factor,invert,range,rational,reciprocal
To divide rational expressions, multiply by the_____ of the divisor.

AG

Ankit Gupta

Numerade Educator

Problem 5

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
f(x)=\frac{2-x}{x-5}
$$

AG

Ankit Gupta

Numerade Educator

Problem 6

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
h(x)=\frac{x-5}{x-2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 7

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
g(x)=\frac{x-3}{(x-2)(x-5)}
$$

AG

Ankit Gupta

Numerade Educator

Problem 8

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
g(x)=\frac{x+3}{(x+2)(x-5)}
$$

AG

Ankit Gupta

Numerade Educator

Problem 9

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
h(x)=\frac{(x-2)(x-3)}{x+3}
$$

AG

Ankit Gupta

Numerade Educator

Problem 10

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
f(x)=\frac{(x+2)(x+3)}{x-3}
$$

AG

Ankit Gupta

Numerade Educator

Problem 11

For each rational function, find the function values indicated, provided the value exists.
$$
f(x)=\frac{2 x^{2}-x-5}{x-1} ;(\text { a) } f(0) ;(\text { b) } f(-1) ; \text { (c) } f(3)
$$

AG

Ankit Gupta

Numerade Educator

Problem 12

For each rational function, find the function values indicated, provided the value exists.
$$
v(t)=\frac{t^{2}+5 t-9}{t+4} ;(\text { a ) } v(0) ;(\text { b) } v(-3) ;(c) v(6)
$$

AG

Ankit Gupta

Numerade Educator

Problem 13

For each rational function, find the function values indicated, provided the value exists.
$$
r(t)=\frac{t^{2}-8 t-9}{t^{2}-4} ;(\text { a ) } r(0) ; \text { (b) } r(2) ; \text { (c) } r(-1)
$$

AG

Ankit Gupta

Numerade Educator

Problem 14

For each rational function, find the function values indicated, provided the value exists.
$$
g(x)=\frac{2 x^{3}-x}{x^{2}-6 x+9} ;(\text { a ) } g(0) ;(\text { b) } g(-2) ;(\text { c) } g(3)
$$

AG

Ankit Gupta

Numerade Educator

Problem 15

Gregory usually takes I hr
longer than Alayna does to prepare the day's soups for Roux Palace. If Alayna takes thr to prepare the day's
soups, the function given by
$$
H(t)=\frac{t^{2}+t}{2 t+1}
$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to prepare the soups if Alayna can prepare them alone in $5 \mathrm{hr} ?$

AG

Ankit Gupta

Numerade Educator

Problem 16

Gregory usually takes I hr
longer than Alayna does to prepare the day's soups for Roux Palace. If Alayna takes thr to prepare the day's
soups, the function given by
$$
H(t)=\frac{t^{2}+t}{2 t+1}
$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to prepare the soups if Alayna can prepare them alone in $2 \mathrm{hr} ?$

AG

Ankit Gupta

Numerade Educator

Problem 17

Simplify by removing a factor equal to 1.
$$
\frac{8 t^{4}}{40 t}
$$

AG

Ankit Gupta

Numerade Educator

Problem 18

Simplify by removing a factor equal to 1.
$$
\frac{35 n}{5 n^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 19

Simplify by removing a factor equal to 1.
$$
\frac{24 x^{3} y}{30 x^{5} y^{8}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 20

Simplify by removing a factor equal to 1.
$$
\frac{10 y z^{4}}{40 y^{2} z^{9}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 21

Simplify by removing a factor equal to 1.
$$
\frac{2 a-10}{2}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 22

Simplify by removing a factor equal to 1.
$$
\frac{3 a+12}{3}
$$

Ben Thatcher

Numerade Educator

Problem 23

Simplify by removing a factor equal to 1.
$$
\frac{5}{25 y-30}
$$

AG

Ankit Gupta

Numerade Educator

Problem 24

Simplify by removing a factor equal to 1.
$$
\frac{21}{6 x-9}
$$

AG

Ankit Gupta

Numerade Educator

Problem 25

Simplify by removing a factor equal to 1.
$$
\frac{3 x-12}{3 x+15}
$$

AG

Ankit Gupta

Numerade Educator

Problem 26

Simplify by removing a factor equal to 1.
$$
\frac{4 y-20}{4 y+12}
$$

AG

Ankit Gupta

Numerade Educator

Problem 27

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(x)=\frac{5 x+30}{x^{2}+6 x}
$$

AG

Ankit Gupta

Numerade Educator

Problem 28

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(x)=\frac{3 x+30}{x^{2}+10 x}
$$

AG

Ankit Gupta

Numerade Educator

Problem 29

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(x)=\frac{x^{2}-9}{5 x+15}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 30

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(x)=\frac{8 x-16}{x^{2}-4}
$$

Ben Thatcher

Numerade Educator

Problem 31

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
h(x)=\frac{2-x}{7 x-14}
$$

AG

Ankit Gupta

Numerade Educator

Problem 32

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
h(x)=\frac{4-x}{12 x-48}
$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(t)=\frac{t^{2}-16}{t^{2}-8 t+16}
$$

AG

Ankit Gupta

Numerade Educator

Problem 34

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(t)=\frac{t^{2}-25}{t^{2}+10 t+25}
$$

AG

Ankit Gupta

Numerade Educator

Problem 35

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(t)=\frac{21-7 t}{3 t-9}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 36

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(t)=\frac{12-6 t}{5 t-10}
$$

Ben Thatcher

Numerade Educator

Problem 37

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
h(t)=\frac{t^{2}+5 t+4}{t^{2}-8 t-9}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 38

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
h(t)=\frac{t^{2}-3 t-4}{t^{2}+9 t+8}
$$

Ben Thatcher

Numerade Educator

Problem 39

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(x)=\frac{9 x^{2}-4}{3 x-2}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 40

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
f(x)=\frac{4 x^{2}-1}{2 x-1}
$$

Ben Thatcher

Numerade Educator

Problem 41

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(t)=\frac{16-t^{2}}{t^{2}-8 t+16}
$$

AG

Ankit Gupta

Numerade Educator

Problem 42

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 5.
$$
g(p)=\frac{25-p^{2}}{p^{2}+10 p+25}
$$

AG

Ankit Gupta

Numerade Educator

Problem 43

Multiply and, if possible, simplify.
$$
\frac{3 y^{3}}{5 z} \cdot \frac{10 z^{4}}{7 y^{6}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 44

Multiply and, if possible, simplify.
$$
\frac{20 y}{9 z^{7}} \cdot \frac{6 z^{4}}{5 y^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 45

Multiply and, if possible, simplify.
$$
\frac{8 x-16}{5 x} \cdot \frac{x^{3}}{5 x-10}
$$

AG

Ankit Gupta

Numerade Educator

Problem 46

Multiply and, if possible, simplify.
$$
\frac{5 t^{3}}{4 t-8} \cdot \frac{6 t-12}{10 t}
$$

AG

Ankit Gupta

Numerade Educator

Problem 47

Multiply and, if possible, simplify.
$$
\frac{y^{2}-9}{y^{2}} \cdot \frac{y^{2}-3 y}{y^{2}-y-6}
$$

AG

Ankit Gupta

Numerade Educator

Problem 48

Multiply and, if possible, simplify.
$$
\frac{y^{2}+10 y+25}{y^{2}-9} \cdot \frac{y^{2}+3 y}{y+5}
$$

AG

Ankit Gupta

Numerade Educator

Problem 49

Multiply and, if possible, simplify.
$$
\frac{7 a-14}{4-a^{2}} \cdot \frac{5 a^{2}+6 a+1}{35 a+7}
$$

Jeffrey Morris

Numerade Educator

Problem 50

Multiply and, if possible, simplify.
$$
\frac{a^{2}-1}{2-5 a} \cdot \frac{15 a-6}{a^{2}+5 a-6}
$$

Jeffrey Morris

Numerade Educator

Problem 51

Multiply and, if possible, simplify.
$$
\frac{t^{3}-4 t}{t-t^{4}} \cdot \frac{t^{4}-t}{4 t-t^{3}}
$$

Jeffrey Morris

Numerade Educator

Problem 52

Multiply and, if possible, simplify.
$$
\frac{x^{2}-6 x+9}{12-4 x} \cdot \frac{x^{6}-9 x^{4}}{x^{3}-3 x^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 53

Multiply and, if possible, simplify.
$$
\frac{c^{3}+8}{c^{5}-4 c^{3}} \cdot \frac{c^{6}-4 c^{5}+4 c^{4}}{c^{2}-2 c+4}
$$

Jeffrey Morris

Numerade Educator

Problem 54

Multiply and, if possible, simplify.
$$
\frac{t^{3}-27}{t^{4}-9 t^{2}} \cdot \frac{t^{5}-6 t^{4}+9 t^{3}}{t^{2}+3 t+9}
$$

AG

Ankit Gupta

Numerade Educator

Problem 55

Multiply and, if possible, simplify.
$$
\frac{a^{3}-b^{3}}{3 a^{2}+9 a b+6 b^{2}} \cdot \frac{a^{2}+2 a b+b^{2}}{a^{2}-b^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 56

Multiply and, if possible, simplify.
$$
\frac{x^{3}+y^{3}}{x^{2}+2 x y-3 y^{2}} \cdot \frac{x^{2}-y^{2}}{3 x^{2}+6 x y+3 y^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 57

Divide and, if possible, simplify.
$$
\frac{12 a^{3}}{5 b^{2}} \div \frac{4 a^{2}}{15 b}
$$

AG

Ankit Gupta

Numerade Educator

Problem 58

Divide and, if possible, simplify.
$$
\frac{9 x^{7}}{8 y} \div \frac{15 x^{2}}{4 y}
$$

AG

Ankit Gupta

Numerade Educator

Problem 59

Divide and, if possible, simplify.
$$
\frac{5 x+20}{x^{6}} \div \frac{x+4}{x^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 60

Divide and, if possible, simplify.
$$
\frac{3 a+15}{a^{9}} \div \frac{a+5}{a^{8}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 61

Divide and, if possible, simplify.
$$
\frac{25 x^{2}-4}{x^{2}-9} \div \frac{2-5 x}{x+3}
$$

Jeffrey Morris

Numerade Educator

Problem 62

Divide and, if possible, simplify.
$$
\frac{4 a^{2}-1}{a^{2}-4} \div \frac{2 a-1}{2-a}
$$

Jeffrey Morris

Numerade Educator

Problem 63

Divide and, if possible, simplify.
$$
\frac{5 y-5 x}{15 y^{3}} \div \frac{x^{2}-y^{2}}{3 x+3 y}
$$

AG

Ankit Gupta

Numerade Educator

Problem 64

Divide and, if possible, simplify.
$$
\frac{x^{2}-y^{2}}{4 x+4 y} \div \frac{3 y-3 x}{12 x^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 65

Divide and, if possible, simplify.
$$
\frac{y^{2}-36}{y^{2}-8 y+16} \div \frac{3 y-18}{y^{2}-y-12}
$$

AG

Ankit Gupta

Numerade Educator

Problem 66

Divide and, if possible, simplify.
$$
\frac{x^{2}-16}{x^{2}-10 x+25} \div \frac{3 x-12}{x^{2}-3 x-10}
$$

AG

Ankit Gupta

Numerade Educator

Problem 67

Divide and, if possible, simplify.
$$
\frac{x^{3}-64}{x^{3}+64} \div \frac{x^{2}-16}{x^{2}-4 x+16}
$$

Jeffrey Morris

Numerade Educator

Problem 68

Divide and, if possible, simplify.
$$
\frac{8 y^{3}-27}{64 y^{3}-1} \div \frac{4 y^{2}-9}{16 y^{2}+4 y+1}
$$

Jeffrey Morris

Numerade Educator

Problem 69

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
f(t)=\frac{t^{2}-100}{5 t+20} \cdot \frac{t+4}{t-10}
$$

AG

Ankit Gupta

Numerade Educator

Problem 70

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
g(n)=\frac{n+5}{n-5} \cdot \frac{n^{2}-25}{2 n+2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 71

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
g(x)=\frac{x^{2}-2 x-35}{2 x^{3}-3 x^{2}} \cdot \frac{4 x^{3}-9 x}{7 x-49}
$$

AG

Ankit Gupta

Numerade Educator

Problem 72

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
h(t)=\frac{t^{2}-10 t+9}{t^{2}-1} \cdot \frac{1-t^{2}}{t^{2}-5 t-36}
$$

AG

Ankit Gupta

Numerade Educator

Problem 73

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
f(x)=\frac{x^{2}-4}{x^{3}} \div \frac{x^{5}-2 x^{4}}{x+4}
$$

AG

Ankit Gupta

Numerade Educator

Problem 74

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
g(x)=\frac{x^{2}-9}{x^{2}} \div \frac{x^{5}+3 x^{4}}{x+2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 75

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
h(n)=\frac{n^{3}+3 n}{n^{2}-9} \div \frac{n^{2}+5 n-14}{n^{2}+4 n-21}
$$

AG

Ankit Gupta

Numerade Educator

Problem 76

Write simplified for each of the following. Be sure to list all restrictions on the domain.
$$
f(x)=\frac{x^{3}+4 x}{x^{2}-16} \div \frac{x^{2}+8 x+15}{x^{2}+x-20}
$$

AG

Ankit Gupta

Numerade Educator

Problem 77

Perform the indicated operations and, if possible, simplify.
$$
\frac{4 x^{2}-9 y^{2}}{8 x^{3}-27 y^{3}} \div \frac{4 x+6 y}{3 x-9 y} \cdot \frac{4 x^{2}+6 x y+9 y^{2}}{4 x^{2}-8 x y+3 y^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 78

Perform the indicated operations and, if possible, simplify.
$$
\frac{5 x^{2}-5 y^{2}}{27 x^{3}+8 y^{3}} \div \frac{x^{2}-2 x y+y^{2}}{9 x^{2}-6 x y+4 y^{2}} \cdot \frac{6 x+4 y}{10 x-15 y}
$$

AG

Ankit Gupta

Numerade Educator

Problem 79

Perform the indicated operations and, if possible, simplify.
$$
\frac{a^{3}-a b^{2}}{2 a^{2}+3 a b+b^{2}} \cdot \frac{4 a^{2}-b^{2}}{a^{2}-2 a b+b^{2}} \div \frac{a^{2}+a}{a-1}
$$

AG

Ankit Gupta

Numerade Educator

Problem 80

Perform the indicated operations and, if possible, simplify.
$$
\frac{2 x+4 y}{2 x^{2}+5 x y+2 y^{2}} \cdot \frac{4 x^{2}-y^{2}}{8 x^{2}-8} \div \frac{x^{2}+4 x y+4 y^{2}}{x^{2}-6 x y+9 y^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 81

Evelyn incorrectly simplifies $\frac{x+2}{x}$ as
$$
\frac{x+2}{x}=\frac{x+2}{x}=1+2=3
$$
She insists that this is correct because it checks when $x$ is replaced with $1 .$ Explain her misconception.

AG

Ankit Gupta

Numerade Educator

Problem 82

Give a step-by-step procedure that a classmate could use to divide rational expressions.

AG

Ankit Gupta

Numerade Educator

Problem 83

Factor completely.
$$2 n^{2}-11 n+12 \quad[5.4]$$

AG

Ankit Gupta

Numerade Educator

Problem 84

Factor completely.
$$
7 y^{2}-28 \quad[5.5]
$$

AG

Ankit Gupta

Numerade Educator

Problem 85

Factor completely.
$$
8 x^{3}+125 \quad[5.6]
$$

AG

Ankit Gupta

Numerade Educator

Problem 86

Factor completely.
$$
\begin{aligned}
&100 a^{2}+60 a b+9 b^{2}\\
&[5.5]
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 87

Factor completely.
$$
t^{3}+8 t^{2}-33 t \quad[5.4]
$$

AG

Ankit Gupta

Numerade Educator

Problem 88

Factor completely.
$$
z^{12}-1 \quad[5.6]
$$

AG

Ankit Gupta

Numerade Educator

Problem 89

Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ

Ben Thatcher

Numerade Educator

Problem 90

Todd incorrectly argues that since
$$
\frac{a^{2}-4}{a-2}=\frac{a^{2}}{a}+\frac{-4}{-2}=a+2
$$
is correct, it follows that
$$
\frac{x^{2}+9}{x+1}=\frac{x^{2}}{x}+\frac{9}{1}=x+9
$$
Explain his misconception.

AG

Ankit Gupta

Numerade Educator

Problem 91

Calculate the slope of the line passing through $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=x^{2}+5 .$ Be sure your answer is simplified.
CAN'T COPY THE GRAPH

Christine Tabor

Christine Tabor

Numerade Educator

Problem 92

Calculate the slope of the line passing through the points $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=3 x^{2} .$ Be sure your answer is simplified.

Ben Thatcher

Numerade Educator

Problem 93

Graph the function given by
$$
f(x)=\frac{x^{2}-9}{x-3}
$$
(Hint: Determine the domain of $f$ and simplify.

Ben Thatcher

Numerade Educator

Problem 94

Perform the indicated operations and simplify.
$$
\frac{r^{2}-4 s^{2}}{r+2 s} \div(r+2 s)^{2}\left(\frac{2 s}{r-2 s}\right)^{2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 95

Perform the indicated operations and simplify.
$$
\frac{d^{2}-d}{d^{2}-6 d+8} \cdot \frac{d-2}{d^{2}+5 d} \div\left(\frac{5 d^{2}}{d^{2}-9 d+20}\right)^{2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 96

Perform the indicated operations and simplify.
$$
\frac{6 t^{2}-26 t+30}{8 t^{2}-15 t-21} \cdot \frac{5 t+15}{t^{2}-4} \div \frac{5 t+15}{t^{2}-4}
$$

AG

Ankit Gupta

Numerade Educator

Problem 97

Simplify.
$$
\frac{m^{2}-t^{2}}{m^{2}+t^{2}+m+t+2 m t}
$$

AG

Ankit Gupta

Numerade Educator

Problem 98

Simplify.
$$
\frac{a^{3}-2 a^{2}+2 a-4}{a^{3}-2 a^{2}-3 a+6}
$$

Christine Tabor

Christine Tabor

Numerade Educator

Problem 99

Simplify.
$$
\frac{x^{3}+x^{2}-y^{3}-y^{2}}{x^{2}-2 x y+y^{2}}
$$

Ben Thatcher

Numerade Educator

Problem 100

Simplify.
$$
\frac{u^{6}+v^{6}+2 u^{3} v^{3}}{u^{3}-v^{3}+u^{2} v-u v^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 101

Simplify.
$$
\frac{x^{5}-x^{3}+x^{2}-1-\left(x^{3}-1\right)(x+1)^{2}}{\left(x^{2}-1\right)^{2}}
$$

AG

Ankit Gupta

Numerade Educator

Problem 102

Let
$g(x)=\frac{2 x+3}{4 x-1}$
Determine each of the following.
a) $g(x+h)$
b) $g(2 x-2) \cdot g(x)$
c) $g\left(\frac{1}{2} x+1\right) \cdot g(x)$

AG

Ankit Gupta

Numerade Educator

Problem 103

Let
$f(x)=\frac{4}{x^{2}-1} \quad$ and $\quad g(x)=\frac{4 x^{2}+8 x+4}{x^{3}-1}$
Find each of the following.
a) $(f \cdot g)(x)$
b) $(f / g)(x)$
c) $(g / f)(x)$

AG

Ankit Gupta

Numerade Educator

Problem 104

Use a graphing calculator to show that
$$
\frac{x^{2}-16}{x+2} \neq x-8
$$

AG

Ankit Gupta

Numerade Educator

Problem 105

To check Example $4,$ Kara graphs
$$
y_{1}=\frac{7 x^{2}+21 x}{14 x} \text { and } y_{2}=\frac{x+3}{2}
$$
since the graphs of $y_{1}$ and $y_{2}$ appear to be identical, Kara believes that the domains of the functions described by $y_{1}$ and $y_{2}$ are the same, $\mathbb{R} .$ How could you convince Kara otherwise?

AG

Ankit Gupta

Numerade Educator