Math 110 Final Exam Review
Revised December 2015
Factor out the GCF from each polynomial.
1) 60x - 15
2) 7x8 y + 42x6
3) x9 y5 - x9 y4 + x7 y2 - x6 y2
Factor each four-term polynomial by grouping.
4) xy + 11x - 6y - 66
5) 6a 3 + 10a 2 b - 9ab2 - 15b3
Factor each trinomial completely. If the polynomial cannot be factored, write "prime."
6) a 2 - 2ab - 35b2
7) 4x2 - 30x + 2x3
8) 2x3 y5 + 2x 2 y5 - 24xy5
9) 5x2 - x - 42
10) 7x4 + 19x2 - 6
11) 36x3 y + 72x2 y + 32xy
12) 25x2 + 40x + 16
Factor each binomial completely.
13) 49x2 - 81
14) x4 - 16
15) 16a 3 - 25a
16) 64y3 - 1
17) 108x3 + 500
18) p3 q3 + 64
1
Solve each equation by factoring.
19) 16y2 + 32y + 15 = 0
20) x(3x + 16) = 12
21) 2(x - 5) = -6x + 2(x 2 - 10)
22) -4x2 - 18x = 4x2 - 2x
23) y3 + 20y2 + 100y = 0
Set up a polynomial equation to model the problem. Then solve.
24) The width of a rectangle is 6 kilometers less than twice its length. If its area is 108 square kilometers, find the
dimensions of the rectangle.
25) An object is thrown upward from the top of a 160-foot building with an initial velocity of 48 feet per second.
The height h of the object after t seconds is given by the quadratic equation h = -16t2 + 48t + 160. When will the
object hit the ground?
26) One leg of a right triangle is 14 inches longer than the smaller leg, and the hypotenuse is 16 inches longer than
the smaller leg. Find the lengths of the sides of the triangle.
Simplify each rational expression.
4x + 2
27)
2
20x + 22x + 6
28)
a 2 - ab + 11a - 11b
a + 11
Perform the indicated operation on rational expressions. Simplify if possible.
x2 + x
56
·
29)
8
x+1
30)
x2 + 8x + 12
x 2 + 4x
·
2
2
x + 10x + 24 x - 3x - 10
31)
p2 - 10p + pq - 10q
p - 10
÷
2
2
2p
- 2q
11p - 11q
32)
3x - 8
2x + 1
2
2
x - 16x + 63 x - 16x + 63
2
33)
5
5
x + 7 9x + 63
34)
x-5
3x + 4
+
2
2
x - 10x + 24 x - 5x + 4
Simplify each complex fraction.
35)
1
k+ 6
5
2
k - 36
36)
4+
2
x
x 1
+
3 6
Solve each rational equation and check the solution.
5-a 3 7
+ =
37)
a
4 a
38)
x
2x - 3
-2x
=
+
2x + 2 4x + 4
x+1
39)
1
2
-1
+
=
x + 4 x + 3 x 2 + 7x + 12
40) 5 -
x
7
=
x-7
7-x
Solve each equation for the indicated variable.
Bh
for B
41) V =
3
42) A =
43) I =
h(a + b)
2
nE
nr + R
for b
for n
3
Set up a rational equation to model the problem. Then solve.
44) Five divided by the difference of a number and 1 equals the quotient of 10 and the sum of the number and 11.
Find the number.
45) A painter can finish painting a house in 7 hours. Her assistant takes 9 hours to finish the same job. How long
would it take for them to complete the job if they were working together?
46) A baker can decorate the day's cookie supply four times as fast as his new assistant. If they decorate all the
cookies working together in 24 minutes, how long would it take for each of them to decorate the cookies
working individually?
47) A cyclist bikes at a constant speed for 21 miles. He then returns home at the same speed but takes a different
route. His return trip takes one hour longer and is 26 miles. Find his speed.
Write an equation for each line described below.
2
48) Slope - ; passing through (2, 3); write in slope-intercept form.
3
49) Passing through (-7, -3) and (0, 5); write in standard form.
50) Undefined slope; passing through (2, -3).
51) Passing through (-4, -1); perpendicular to x + 3y = -3; write in slope-intercept form.
52) Passing through (6, 10); parallel to 8x + 7y = 90; write in slope-intercept form.
53) Passing through (-7, 5); perpendicular to x = 3.
54) The average value of a certain type of automobile was $14,460 in 1,994 and depreciated to $8,340 in 1,998. Let y
be the average value of the automobile in the year x, where x = 0 represents 1,994. Write a linear equation that
models the value of the automobile in terms of the year x.
55) Write the equation of the line graphed below in slope-intercept form.
4
Determine whether the relation is also a function.
56) x = 4y2
Find the domain and range of each relation. Also, state whether the relation is a function.
57) {(-2, 8), (-1, -7), (-3, -9), (-3, -1)}
58)
59)
60)
5
Find the domain and range of each relation. Also, state whether the relation is a function.
61) pets at home
Alice
Brad
Carl
cat
dog
Given the following function, find the indicated values.
62) Find h(-1), h(0), and h(-4) when h(x) = x2 - 2x - 3.
63) If f(x) =
x3 + 5
, find f(-3), f(3), and f(5)
x2 + 5
Find the domain of the rational function. Write the answer in set-builder notation and interval notation.
4x 2 - 9
64) f (x) =
7x - 35
65) f(x) =
1 - 6x
x3 - 6x2 - 27x
Graph each linear function.
66) f(x) = 4x - 5
67) f(x) = 3x
6
Graph each piecewise defined function.
68)
f(x) = 1x if x 1
x - 3 if x < 1
69)
f(x) =
3x + 2 if x 2
-x if x > 2
Simplify each radical. Assume that all variables represent nonnegative real numbers.
70) 16x10
71)
3
72) -
73)
-8x6
4
81
4 16x4
y12
Evaluate the radical function.
3
74) If f(x) = x + 21, find the value of f(6).
7
Convert the expression into radical notation. Then simplify if possible.
75) (-32)1/5
76) -361/2
77) (-27)4/3
78) 16-3/2
Use the properties of exponents to simplify each expression. Write with positive exponents.
y3/4
79)
y1/4
80) z -2/7 · z 3/7
2
81) (r1/5 · s1/5 )
Simplify each radical expression. Assume that all variables represent positive real numbers.
82) 28 · 7
83)
96x2 y
84)
80
49
85)
90x 7
10x
86)
87)
88)
5
3
6
243 x4 y17
-27a 11b13
x15y15
Perform the indicated operation. Simplify, if possible. Assume all variables represent positive real numbers.
89) -5 150 + 4 24 - 6 96
90) 7
3
x3 y13 + 3xy
3
8y10
8
91) 6 3( 11 +
11)2
92) ( 5 -
93) (4 +
3
3)
2)(4 -
3
2)
Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
3
94)
11
95)
13
3
2
96)
7
27x
97)
5
9- 3
Solve each radical equation.
98) 4x - 1 - 5 = 0
99)
3
6x = -4
100) x 101)
20x + 20 = -6
4x + 1 = 3 +
x-2
Write in terms of i.
102) -108
Multiply or divide. Simplify if possible.
103) -9 · -10
104)
-18
3
Perform the indicated operation. Simplify if possible. Write the result in the form a + bi.
105) (6 + 9i) - (-8 + i)
106) (6 + 6i) + (3 - 6i)
9
107) 5i(7 - 4i)
108) (8 + 6i)2
109)
6 - 28i
-7i
110)
9
1 - 2i
111)
7 + 5i
6 - 7i
Use the square root property to solve each equation.
112) x2 = 180
113) 5x2 + 35 = 0
114) (x + 2)2 = 28
115) (2x + 3)2 = 25
Solve each equation by completing the square.
116) x2 + 12x + 22 = 0
117) x2 + 70 = -18x
118) x2 + 10x + 29 = 0
Use the quadratic formula to solve each equation.
119) 2x2 + 10x = - 7
120) 4x2 + 3x + 6 = 0
121)
1 2 1
1
x + x- =0
2
4
2
Solve each equation using a method of your choice.
122) x4 + 21x2 - 100 = 0
123)
2
x2 - 17x + 72
=
2x
x
x-9 x-8
10
Solve.
124) x2/3 - 5x1/3 + 6 = 0
125) x -
5x = 1
Use the Pythagorean theorem to find the unknown side of the right triangle.
126)
2
151
Set up an appropriate equation to model the problem. Then solve.
Solve.
127) A rectangular sign must have an area of 29 square yards. Its length must be 6 yards more than its width. Find
the exact dimensions of the sign.
128) A rocket is launched from the top of a cliff that is 112 feet high with an initial velocity of 280 feet per second.
The height, h, of the rocket after t seconds is given by the equation h = -16t2 + 280t + 112. How long after the
rocket is launched will it strike the ground? Round to the nearest tenth of a second, if necessary.
129) A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut
while the balloon is directly above a flag pole that is 50 feet from where the rope is staked down. Find the
height of the balloon if the rope is 70 feet long.
130) Suppose that an open box is to be made from a square sheet of cardboard by cutting out 5-inch squares from
each corner as shown and then folding along the dotted lines. If the box is to have a volume of 80 cubic inches,
find the original dimensions of the sheet of cardboard.
11
131) Shelly can cut a lawn with a riding mower in 5 hours less time than it takes William to cut the lawn with a push
mower. If they can cut the lawn in 4 hours working together find how long to the nearest tenth of an hour it
takes for William to cut the lawn alone.
Sketch the graph of each quadratic function. Give the vertex and axis of symmetry.
132) f(x) = x2 + 5
133) f(x) = (x - 3)2
134) f(x) = (x + 1)2 - 9
12
135) f(x) = -2(x - 3)2 + 5
Find the vertex of the graph of the quadratic function. Find any intercepts and graph the function.
136) f(x) = -2x2 + 24x - 70
137) f(x) = x2 + 4x - 7
Solve each problem.
138) The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function
C(x) = 3x 2 - 12x + 32. Find the number of automobiles that must be produced to minimize the cost.
139) An arrow is fired into the air with an initial velocity of 128 feet per second. The height in feet of the arrow t
seconds after it was shot into the air is given by the function h(t) = -16t2 + 128t. Find the maximum height of the
arrow.
13
Answer Key
Testname: MATH 110 FINAL EXAM REVIEW - DEC2015
1) 15(4x - 1)
2) 7x6 (x2 y + 6)
3) x6 y2 (x3 y3 - x3 y2 + x - 1)
4) (y + 11)(x - 6)
5) (2a 2 - 3b2 )(3a + 5b)
6) (a + 5b)(a - 7b)
7) 2x(x - 3)(x + 5)
8) 2xy5 (x - 3)(x + 4)
9) (5x + 14)(x - 3)
10) (7x2 - 2)(x2 + 3)
11) 4xy(3x + 4)(3x + 2)
12) (5x + 4)2
13) (7x + 9)(7x - 9)
14) (x2 + 4)(x + 2)(x - 2)
15) a(4a + 5)(4a - 5)
16) (4y - 1)(16y2 + 4y + 1)
17) 4(3x + 5)(9x2 - 15x + 25)
18) (pq + 4)(p2 q2 - 4pq + 16)
19) 20)
3
5
,4
4
2
, -6
3
21)
22)
23)
24)
25)
26)
-1, 5
0, - 2
0, -10
length = 9 km, width = 12 km
5 sec
10 in., 24 in., 26 in.
1
27)
5x + 3
28) a - b
29) 7x
x
30)
x-5
2
31)
11
1
32)
x-7
40
33)
9(x + 7)
34)
4x2 - 20x - 19
(x - 4)(x - 6)(x - 1)
35)
k- 6
5
36)
65) set-builder: {x|x is a real number
and x 9, x -3, x 0}
interval: ( , -3) (-3, 0) (0,
9) (9, )
66)
12
x
37)
38)
39)
40)
-8
3
no solution
No solution
3V
41) B =
h
42) b =
2A - ha
h
43) n =
IR
E - Ir
44) 13
15
hours
45) 3
16
46) baker: 30 minutes
assistant: 120 minutes
47) 5 mph
2
13
48) y = - x +
3
3
67)
49) 8x - 7y = -35
50) x = 2
51) y = 3x + 11
8
118
52) y = - x +
7
7
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
y=5
y = -1,530x + 14,460
y = - 3x - 12
Not a function
Domain: {-1, -2, -3}; range: {-7,
8, -9, -1}; Not a function
Domain: {-1, 0, 1, 2}; range: {5, 3,
-2, -1}; it is a function
Domain: (- , ); range: [0, 5]; it
is a function
Domain: (- , ); range: (- , -1);
it is a function
Domain: {Alice, Brad, Carl};
range: {cat, dog}; it is a function
h(-1) = 0, h(0) = -3, h(-4) = 21
11
16
, f(3) =
, f(5) =
f(-3) = 7
7
13
3
64) set-builder: {x|x is a real
number and x 5}
interval: ( , 5) (5, )
14
68)
Answer Key
Testname: MATH 110 FINAL EXAM REVIEW - DEC2015
69)
95)
13
71) -2x2
72) -3
2x
73)
y3
74)
75)
76)
77)
f(6) = 3
-2
-6
81
1
78)
64
79) y1/2
80) z 1/7
r2/5 s2/5
14
4x 6y
4 5
84)
7
81)
82)
83)
85) 3x3
86) 3y3
5
x4 y2
3
87) -3a 3 b4 a 2 b
88) x5 y5
89) -41 6
90)
91)
92)
93)
94)
3
13 xy4 y
6 33 + 18
16 - 2 55
3
16 - 4
33
11
2
126)
127)
128)
129)
130)
131)
132)
4
96)
7 3x
9x
97)
45 + 5 3
78
98)
13
2
99) 70) 4x5
3
7 3
3 + 38 yards by -3 + 38 yards
17.9 sec
20 6 ft
14 in. by 14 in.
11.2 hours
Vertex: (0, 5); axis: x = 0
32
3
100)
101)
102)
103)
104)
105)
106)
107)
108)
4
2, 6
6i 3
-3 10
i 6
14 + 8i
9
20 + 35i
28 + 96i
6
109) 4 + i
7
110)
9 18
i
+
5
5
111)
7
79
i
+
85 85
133) Vertex: (3, 0); axis: x = 3
{-6 5, 6 5}
{-i 7, i 7}
{-2 - 2 7, -2 + 2 7}
{1, -4}
{-6 - 14, -6 + 14}
{-9 - 11, -9 + 11}
{-5 + 2i, -5 - 2i}
-5 - 11 -5 + 11
,
119)
2
2
112)
113)
114)
115)
116)
117)
118)
120)
-3 - i 87 -3 + i 87
,
8
8
121)
-1 - 17 -1 + 17
,
4
4
122) {-2, 2, -5i, 5i}
7 - 57 7 + 57
,
123)
2
2
124) {8, 27}
7+3 5
125)
2
15
134) Vertex: (- 1, - 9); axis: x = - 1
Answer Key
Testname: MATH 110 FINAL EXAM REVIEW - DEC2015
135) Vertex: (3, 5); axis: x = 3
136) Vertex: (6, 2);
x-intercepts: (5, 0) and (7, 0);
y-intercept: (0, - 70)
137) Vertex: (-2, -11);
x-intercepts: (-2 ± 11, 0);
y-intercept: (0, -7)
138) 2 thousand automobiles
139) 256 ft
16