[PDF] Topic
: Evaluating and Simplifying Algebraic Expressions

101369_6fa18_mat105_final_exam_review_121.pdf

John Jay College of Criminal Justice

The City University of New York

Department of Mathematics and Computer Science

MAT 105 - College Algebra

Departmental Final Examination Review

Topic #1: Evaluating and Simplifying Algebraic Expressions Evaluate the algebraic expression for the given value or values of the variable(s). 1) y - 7x

6x + xy;x =

-2 and y = 3 1) - 17 18 11 6 1 18 2) -b + b2 - 4ac

2a when a =

5, b =

14, and c =

-3 2) 1 5 1 5 -3 D) 3

Simplify the algebraic expressions:

3) (12y + 9) + 11y2 - 6y + 9) 3) A) 11y2 + 18y - 18 B) 29
6 C) - 6y + 18 y+ 18 4) (3a - 2b - 5c) - (9a - 6b - 7c) 4)

6a + 4b + 2c

12a - 8b - 12c

- 12c - 8 5) (x -

11)(x2

+ 7x - 5) 5) x3 - 4x2 - 82x +
55
+ 18x2 + 82x +
55
72
- 72
- 6) -35x2 + 28x + 21 7 6) -5x2 + 4x + 3 35x2
+ 28x + 3 245x2
+ 196 147
28x + 21
7)

20x9y11z9

4x4y3z8

7)

5x4y7z

5 8 x5y8z 1 8)

25x13y6

5x3y3 0

8) A) x10y3 B)

5x10y3

C) 1 D) 0

Topic #2: Integer Exponents

Simplify the exponential expressions:

9) (-6x4)(8x7) 9) -48x28 11 48x11
28
10) (-5x5y-6)(2x-1y) 10) -10x6 y7 3x4 y5 4 5 10 4y7 11)

21x13y13

7x12y-10

11) 3 y3 25
23
23
21
23

Topic #3: Rational Exponents and Radicals

Evaluate the expression :

12) 144
+ 25
12) 13 169
17 119

Add or subtract terms whenever possible.

13) 52
+ 550 13) 102
30
-302 20 14) 2x 68x
- 232x 14) 542x
4 2x

Rationalize the denominator.

15) 3 7 - 2 15) 21
+ 32 5 3 7 - 3 2 - 47

Simplify the radical expression.

16) 3x8 16) x23x x3x x23x2 x3x2 2

Evaluate the expressions :

17) 161/4
17) 8 16 32
2 18) -3/2 18) 1 343
-343 343
- 1 343

Simplify by reducing the index of the radical.

19) 20x16 19) 4x4 5x 5 4 20) 816x4
20) 22x
1 4x 2x 2x

Topic #4: Factoring

Factor out the greatest common factor.

21)
21x4
- 6x3 + 15x2 21)
A) 3(7x4 - 2x3 + 5x2) B) x2(21x2 - 6x + 15) C) x(7x3 - 2x2 5x) D) x2(7x2 - 2x + 5)

Factor by grouping.

22)
x3 + 9x - 3x2 - 27 22)
(x - 3)(x2 + 9) (x - 3)(x + 9) - + Factor the trinomial, or state that the trinomial is prime. 23)
x2 - 12x + 27 23)
+ 9 - 3 1) prime 24)
6x2 + 13x + 6 24)
(6x + 2)(x + 3) 3 - )(2x - 3) +

Factor the difference of two squares.

25)
49x2
- 16y2 25)
(7x + 4y)2 (7x + 4y)(7x - 4y) - prime 3 Factor using the formula for the sum or difference of two cubes. 26)
64x3
- 1 26)
A) (4x + 1)(16x2 - 4x + 1) B) - + C) 1) D) prime 27)
125x3
+ 1 27)
5 25
5

Topic #5: Rational Expressions

Perform the indicated operations and simplify the result. Leave the answer in factored form. 28)

4x - 4

x · 8x2

5x - 5

28)
20x2 + 40x + 20 8x3 32x3
- 32x2 5x2 - 5x 5 32x
32x
5 29)
x2 - 10x + 24
x2 - 21x +
108
· x2 - 14x + 24
x2 - 16x + 60
29)
(x -

4)(x -

2) (x -

9)(x -

10) + + + + (x2 - 10x +

24)(x2

- 14x + 24)
(x2 - 21x +

108)(x2

- 16x + 60)
) 10)

Add or subtract as indicated.

30)
4 x2 - 3x + 2 + 5 x2 - 1 30)
6x 9 1)(x 1)(x - 2)

9x - 6

2) 9 6

40x - 6

31)
x x2 - 16 - 5 x2 + 5x + 4 31)
x2 - 4 (x - 4)(x + 4)(x + 1) x2 + 4x + 20 - x2 - 4x + 20 4) 4

Topic #6: Complex Numbers

Add or subtract as indicated and write the result in standard form. 32)
-7 - (- 2 - 7i) - (- 2 + 5i) 32)
A) 4 - 2i B) -3 - 2i C) + D) + Find the product and write the result in standard form. 33)
(-3 - 7i)(2 + i) 33)
1 17i 11 13 + 11i - 17

Divide and express the result in standard form.

34)
8 4 + i 34)
32
15 + 8 15i 17 - 17 35)
6 - 6i 8 2i 35)
60
36
1 - 1 4i 3 20 1 4i 9 15 Perform the indicated operations and write the result in standard form. 36)
-16 + -81 36)
36i
i 13 37)
-2 - -24 2 37)
-1 - i2 1 + i6 + 6 Topic #7: Linear, Rational, Radical, Absolute Value, and Literal Equations

Solve and check the linear equations.

38)

5x + 4) - 5

= -4(x - 7) 38)
{19} - 29} 6} 29
39)
2x 5 = x 3 + 5 39)
75}
150
75
150}
First, write the value(s) that make the denominator(s) zero. Then solve the equation. 40)
10 x 5 2x + 30
40)
x J 0, 2; 25 6 0; 1 4

No restrictions; {2}

x J 0; {4} 5 Solve the absolute value equation or indicate that the equation has no solution. 41)

3x - 3

= 18 41)
A) {3, -9} B) {9, 3} C) {3} D) + Solve the radical equation, and check all proposed solutions. 42)
6x + 55
= x 42)
- 11 {11} -5, 11}

Solve the formula for the specified variable.

43)
F = 9

5C + 32 for C

43)
C = 5

F - 32

F - 32

9 9

5(F - 32)

5 9 44)
A = 1

2bh for b

44)
b = h 2A 2A h A 2h Ah 2 45)

S = 2"rh + 2"r2 for h

45)
h = S 2"r - 1 h = S - r 2"(S - r) h =

S - 2"r2

2"r 46)

P = 2L + 2W for W

46)
W = P - L 2 2L 2 W = P - L d 2L 6 Topic #8: Linear, Compound, and Absolute Value Inequalities

Solve the linear inequality. Other than +, use interval notation to express the solution set and graph the solution set

on a number line. 47)

7x - 6

L

6x - 2

47)
A) [4, Q) B) (-8, Q) C) Q, 4] D) 48)
-8x + 4 K -2(3x + 1) 48)
3 3 7

Solve the compound inequality. Other than +, use interval notation to express the solution set and graph the

solution set on a number line. 49)
17 K

5x - 3

K 22
49)
A) (4, 5) B) -5, -4) C) [ ] D) ]

Solve the absolute value inequality. Other than +, use interval notation to express the solution set and graph the

50)
|x + 2| + 6 K 11 50)
7 11] 3) ]

Q, -7] 1 [3, Q)

8 51)
|7x - 9| - 3 > -6 51)
A) (-Q, Q) B) 6 7, 12 7 C) Q D) + 52)
5 8 9 < 14 52)
3 5 13 5 -Q, 3 5 13 5

Topic #9: Distance and Midpoint Formulas; Circles

Find the distance between the pair of points.

53)
(-1, 4) and (-5, 7) 53)
6 25
10 5 9 Find the midpoint of the line segment whose end points are given. 54)
(7, 3) and (4, 1) 54)
A) (11, 4) B) 3, 2) C) (11 2, 2) D) 3 2, 1) Write the standard form of the equation of the circle with the given center and radius. 55)
-4, 4); 3 55)
(x - 4)2 + (y + 4)2 = 9 + - 3

Find the center and the radius of the circle.

56)
(x - 5)2 + (y + 7)2 = 36
56)
-5, 7), r = 36
7 -5 7 6 5

Complete the square and write the equation in standard form. Then give the center and radius of the circle.

57)
x2 - 12x + 36 + y - 8y + 16 = 16 57)
(x - 6)2 + (y - 4)2 = 16 (6, 4), r = 4 -6, -4), r = 16 - 4 - 6 -4, -6 4 6

Graph the equation.

58)
(x - 1)2 + (y - 2)2 = 49
58)

Domain = (-6, 8), Range = (-5, 9)

8 6 9 5 10

Topic #10: Basics of Functions and Their Graphs

Determine whether the relation is a function.

59)
{(-7, -1), (-7, 2), (-1, 8), (3, 3), (10, -7)} 59)
A)

Not a function

B)

Function

Evaluate the function at the given value of the independent variable and simplify. 60)
f(x) = -3x 8;f(-2) 60)
22
-2 C) 14 D) 11 61)
f(x) = x +

11;f(-2)

61)
-3 3 1.73 not a real number

Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.

62)
62)
function not a function 63)
63)
11 64)
64)
A) function B) not a function Use the graph to find the indicated function value. 65)
y = f(x). Find f(-1) 65)
-0.2 4.2 C) 4.2 D) 0.2 12 Use the graph to determine the function's domain and range. 66)
66)
A) domain: (-Q, Q) range: [-4, Q) B) range: ( Q, Q) C) domain: [-1, Q) range: [ D) -1) or (-1, Q) -4) or (-4, Q) 67)
67)
domain: [0, Q) range: [0, Q) domain: [0, range: [ 1 ( 13 Identify the intervals where the function is changing as requested. 68)

Increasing

68)
A) (-3, 3) B) 3, Q) C) 2, D) 2, 2) 69)

Constant

69)
1, 1) (2, Q) -1) (1, 2) Evaluate the piecewise function at the given value of the independent variable. 70)
f(x) =3x + 1 if x < -1 -2x - 5 if x L -1; f(2) 70)
-8 9 1 3 Determine whether the given function is even, odd, or neither. 71)
f(x) = x3 - 5x 71)

Neither

Even Odd 72)
2x2 + x4 72)
73)
x2 73)
14

Topic #11: Slope and Linear Functions

Find the slope of the line that goes through the given points. 74)
(-2, -6), (-9, -17) 74)
A) 11 7 B) - 11 7 C) 7 11 D) 23
Use the given conditions to write an equation for the line in point-slope form. 75)

Slope =

4, passing through (-3, 7)

75)
x - 7 =

4(y + 3)

y =

4x + 19

+ 7 =

4(x - 3)

- + Use the given conditions to write an equation for the line in slope intercept form. 76)

Slope =

2

3, passing through (7, 3)

76)
y = 2

3x + 7

- 5 3 mx - 5 3 77)

Passing through (-8, -2) and (-5, -7)

77)
y = - 5 3x - 46
3 + 2 = - 5

3(x + 8)

mx - 46
3 5 3x 46
3 15

Graph the line whose equation is given.

78)
y =

2x - 2

78)
A) B) C) D) Determine the slope and the y-intercept of the graph of the equation. 79)
7x -

10y - 70

= 0 79)
m = 10

7; (0, 10)

m =

7; (0, 70)

7 10 -7) - 7

10; (0, 7)

Use the given conditions to write an equation for the line in the indicated form. 80)
Passing through (2, 3) and parallel to the line whose equation is y = -2x + 3 ; point-slope form 80)
y - 2 = -2(x - 3) 3 2 = 2x x - 2 16 81)
Passing through (5, 3) and perpendicular to the line whose equation is y =

2x + 7;

point-slope form 81)
A) y - 3 = 1 2(x + 5) B) - 1 2(x - 5) C) 5 - 3 D) y = - 2x - 11 Find the average rate of change of the function from x1 to x2. 82)
f(x) = -3x2 - x from x1 = 5 to x2 = 6 82)
- 1 6 -34 2 1 2 Find and simplify the difference quotient of f, f(x + h) - f(x) h, hJ 0, for the function. 83)
f(x) = 4x2 83)
4 4(2x2 + 2xh + h2) h (2x+h) 8 h + x + 4h 17

Topic #12: Transformations of Graphs

Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the given function. 84)
h(x) = (x - 7)2 - 5 84)
A) B) C) D) 18

Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g.

85)
g(x) = -f(x - 1) + 2 y = f(x) 85)
A) B) C) D) Topic #13: Algebra of Functions, Function Composition, and Inverse Functions Given functions f and g, perform the indicated operations. 86)
f(x) = 3 - 5x,g(x) = -8x + 5

Find f + g.

86)
-5x

3x + 8

8x + 3

13x + 8

For the given functions f and g , find the indicated composition. 87)
x + 9,g(x) =

5x - 1

(fHg)(x) 87)

15x + 8

44
6 12 19 88)
f(x) = x2 + 2x + 2,g(x) = x2 - 2x - 3 (fHg)(-3) 88)
A) 51
B) 136
C) 170
D) 17

The functio is one-to-one. Find its inverse.

89)
f(x) =

3x - 7

89)
f-1(x) = x + 7 3 x 3 - 7 + - 90)
f(x) = x + 7 90)
f-1(x) = x2 + 7, x L 0 (x + 7)2 f-1(x) = x2 - 7, x L 0 x - 7 91)
f(x) =

3x - 7

8x + 4

91)
f-1(x) = -4x - 7

8x - 3

3x - 7

8x + 4

8x - 3

-4x - 7 + 3 Topic #14: Quadratic Equations and Quadratic Functions

Solve the equation by factoring.

92)
x2 = x + 6 92)
{-2, 3} {1, 6} -3} 2, 3

Solve the equation by factoring.

93)
+ 2x - 120 = 0 93)

12, -10}

12, 1}

10} ,

Solve the equation by the square root property.

94)
6x2 = 54
94)
{-36, 36} 6 6 3 {0} 95)
(x - 3)2 = 49
95)
52}
10 -4} 7 7 4 10}

Solve the equation by completing the square.

96)
14x + 26 = 0 96)
{-7 -

23 , -7

+ 23}
7 -

26 , 7

+ 26}
14 + 26}
+ 23}
20

Solve the equation using the quadratic formula.

97)
x2 + 7x + 7 = 0 97)
A) -7 - 21

14, -7

+ 21
14 B) 2 2 C) 77
2, -7 + 77
2 D) 7 - 21
2, 7 + 21
2 98)
5x2 - 3x + 3 = 0 98)
3

± i51

10 -3 ± 51
10 i51 10 51
10 The graph of a quadratic function is given. Determine the function's equation. 99)
99)
h(x) = (x - 2)2 + 2 g(x) + - j(x) = (x - 2)2 - 2 f(x) = (x + 2)2 + 2 100)
100)
-x2 - 2x - 1 -x2 + 2x + 1 -x2 + 1 - 1 Find the coordinates of the vertex for the parabola defined by the given quadratic function. 101)
f(x) = (x - 4)2 - 4 101)
(0, -4) (4, 4) -4, 0) 21
102)
y + 4 = (x - 2)2 102)
A) (2, - 4) B) - 2, - 4) C) 4 2 D) 2) Find the axis of symmetry of the parabola defined by the given quadratic function. 103)
f(x) = x2 + 7 103)
x = -7 7 0 y = 7 104)
(x + 4)2 - 6 104)
6 6 4 4 Topic #15: Introduction to Polynomial and Rational Functions Form a polynomial whose zeros and degree are given. 105)

Zeros: -3, -2, 2; degree 3

105)
f(x) = x3 - 3x2 + 4x - 12 for a = 1 + + -

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the

x-axis at each x -intercept. 106)
f(x) =

5(x + 3)(x - 3)3

106)
-3, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis

3, multiplicity 1, touches x-axis; -3, multiplicity 3

, multiplicity 1, crosses x-axis; - , multiplicity 1, touches x-axis; 107)
2(x2 + 4)(x + 1)2 107)

1, multiplicity 2, touches x

axis , multiplicity 2, crosses x axis 4 -1, multiplicity 2, touches x-axis -1, multiplicity 2, crosses x

Find the x- and y-intercepts of f.

108)
f(x) = (x +

4)(x -

2)(x +

2) 108)
x-intercepts: -4, -2, 2; y-intercept: -16 2

2, 4; y-intercept: -16

16 109)
4x - x3 109)
intercepts: 0, -4; y-intercept: 0

2, -2; y-intercept: 0

intercept: 4 intercept: 4 List the potential rational zeros of the polynomial function. Do not find the zeros. 110)
6x4 + 2x3 3x2 + 2 110)
± 1

6, ±

1

3, ±

1

2, ±

2

3, ± 1, ± 2, ± 3

1, ± 2 2 3 2 1, ± 2, ± 3, ± 6 22
Use the Remainder Theorem to find the remainder when f(x) is divided by x - c. 111)
f(x) = x4 + 8x3 +

12x2; x + 1

111)
A)

R = 21

B) -21 C) 5 D) 5 Form a polynomial f(x) with real coefficients having the given degree and zeros. 112)

Degree 3: zeros: 1 + i and -5

112)
f(x) = x3 + x2 - 8x + 10 -5x2 - 8x - 12 3x2 - 8x + 10 + 10x - 8 Use the given zero to find the remaining zeros of the function. 113)
f(x) = x4 - 21x2 - 100; zero: -2i 113)

2i, 5i, -5i

10, -10

i, -10i , -5

Divide using synthetic division.

114)
x4 - 3x3 + x2 + 4x - 5 x - 1 114)
x3 + 2x2 - x + 5 - 2 x - 1 - + x + 3 + 4 x - 1 + + 4 3 Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 115)
3x4 + 4x3 - 4x2 + 3x - 2 115)
rises to the left and rises to the right falls to the left and rises to the right falls to the left and falls to the right rises to the left and falls to the right 116)
2 3 + 5 2 + 5x + 5 116)

Find the domain of the rational function.

117)
g(x) = 2x x + 2 117)
{x|x J -2} all real numbers 2} 0} 118)
f(x) = x + 7 x2 - 9 118)

3, x J

3, x J

-7} } 0, x J 9} 119)
x + 2 + 16x 119)
-16} 4 4 2 23
Find the vertical asymptotes of the rational function. 120)
h(x) = 4x2 (x + 2)(x - 6) 120)
A) x = -2, x = 6 B) , x = -4 C)

2, x =

- D) 4 121)
g(x) = x + 4 x2 + 4 121)
2 none -4 24

Answer Key

Testname: MAT105_FINAL EXAM_REVIEW 121

1) A 2) 3) D 4) 5) 6) 7) C 8) 9) B 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
22)
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25

Answer Key

Testname: MAT105_FINAL EXAM_REVIEW 121

49)
C 50)
51)
A 52)
D 53)
54)
55)
B 56)
57)
58)
59)
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26

Answer Key

Testname: MAT105_FINAL EXAM_REVIEW 121

97)
B 98)
A 99)
D 100)
C 101)
102)
103)
104)
105)
106)
107)
108)
109)
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