MAT 102 Practice Final Exam
This review is intended to cover many different types of problems in this class. Work your
problems on a separate piece of paper and check against the answer sheet. You need to do other
things to prepare for this exam such as:
reviewing all of your unit tests,
certifying in any sections that you have missed,
reworking Hawkes homework problems in any section that you have troubles with, and
creating/taking multiple practice tests through Hawkes.
---------------------------------------------------------------------------------------------------------Factorization
Factor completely. If it cannot be factored, write "Not Factorable".
1.
18𝑏𝑞 − 8𝑢𝑦 + 3𝑞𝑢 − 48𝑏𝑦
2.
48𝑥 3 𝑦 − 44𝑥 2 𝑦 + 8𝑥𝑦
3.
5.
4.
4𝑥 2 − 25𝑦 2
Solve this equation using zero-factor property:
27𝑦 3 + 8
𝑥 3 − 4𝑥 2 = 5𝑥
------------------------------------------------------------------------------------------------Rational expressions
6.
Determine for which values of 𝑥 the following rational expression is undefined:
𝑥+13
𝑥 2 −3𝑥−10
7.
𝑥−4
Reduce the following rational expression to its lowest terms.
4−𝑥
Perform the indicated operations on the rational expressions and simplify.
8.
𝑥𝑦
𝑥2
13.
𝑥
2𝑥 2 +𝑥−15
𝑥+4
5𝑥
4𝑥 − 5
4𝑥2
10.
12.
+ 8𝑥 + 15
÷
2
𝑥+3
9.
11.
−
3
7𝑥
÷
8
𝑥+1
+
−3𝑥 − 11
𝑥 2 + 3𝑥 + 2
3𝑥 −2 + 5𝑦 −3
2𝑥 −3 − 3𝑦 −2
𝑥+3
35𝑥
For the following equation, state the restriction(s) on the variable. Then solve.
𝑥
𝑥+1
−
𝑥
𝑥2
+ 3𝑥 + 2
=
2
𝑥+2
14.
The denominator of a fraction is four more than the numerator. If both numerator and
4
denominator are decreased by eight, the simplified result is 5 . Find the original fraction.
(Do NOT simplify.)
15.
𝑧 varies directly as √𝑥 and inversely as 𝑦 2 . If 𝑧 = 64 when 𝑥 = 16 and 𝑦 = 8 , find 𝑧
if 𝑥 = 25 and = 7 . (Round off your answer to the nearest hundredth.)
--------------------------------------------------------------------Radical and Rational Exponent Expressions
Simplify by performing the indicated operations. Rationalize any denominators. Assume that each
variable is positive.
16.
√8𝑥 7 𝑦10 𝑧
17.
18.
√4𝑦 + 4√25𝑦 − 8√𝑦
19.
20.
𝑦
21.
√2𝑥
3
√
38𝑥 10
8𝑦 9
(√3𝑥 + 4)(√3𝑥 + 3)
√𝑦
√𝑦 − √3
Use the rules of exponents to simplify the following expression. Assume all variables are positive.
Write the solution using rational exponents.
22.
27𝑥 · 𝑦 −2
1⁄
3
(𝑥 −1 · 𝑦−1 )
Change to an equivalent expression in exponential notation, and then simplify. Assume that the
variable is positive.
23.
√𝑦 3
3
√𝑦 2
Solve each of the following radical equations. Remember to check your solution(s).
24. √−1 − 𝑣 + 3 = 6
25. √3 + 13𝑦 = 𝑦 + 3
---------------------------------------------------------------------------------------------------Complex Numbers
Perform the indicated operations. Write your final answer in standard form.
(2 − 5𝑖) − (−4 − 6𝑖) − (4 + 6𝑖)
26. (3 + √−49) + (−8 + √−1)
27.
28. (−4 + 8𝑖)(1 − 5𝑖)
------------------------------------------------------------------------------------------------Quadratic Equations
Solve each of the following quadratic equations by the indicated method. Simplify your answers
and rationalize denominators if necessary.
29. 2 (𝑦 + 4)2 = 54
by the square root method
30.
2𝑥 2 + 12𝑥 = −8
by completing the square method
31.
−3𝑥 2 − 2𝑥 − 3 = 0
by the quadratic formula method
Solve each of the following equations. Simplify your answers and rationalize denominators if
necessary.
32. 𝑥 −2 − 16𝑥 −1 + 28 = 0
33.
𝑦 5 − 144𝑦 = 0
2
Solve each of the following. To receive full credit, show:
(a) what your variable represents
(b) an equation
(c) solving the equation
(d) the answer to the question
34.
One positive integer is 6 less than twice another. The sum of their squares is 569.
Find the integers.
35.
A rectangular auditorium seats 1200 people. The number of seats in each row
exceeds the number of rows by 10. Find the number of seats in each row.
36.
An airplane can travel 310 mph in still air. If it travels 1770 miles with the wind in
the same length of time it travels 1330 miles against the wind, what is the speed of
the wind?
37.
An inlet pipe on a swimming pool can be used to fill the pool in 10 hours. The drain
1
pipe can be used to empty the pool in 20 hours. If the pool is 5 filled and then the
inlet pipe and drain pipe are opened, how long from that time will it take to fill the
pool?
38. An arrow is shot vertically upward from a platform 21 ft high at a rate of 188 ft per sec.
When will the arrow hit the ground? Use the formula: ℎ = −16𝑡 2 + 𝑣0 𝑡 + ℎ0 . (Round off your
answer to the nearest tenth.)
39.
A ball is thrown vertically upward from the ground with an initial velocity of 109 ft per sec.
Use ℎ = −16𝑡 2 + 𝑣0 𝑡 + ℎ0 .
Step 1. Determine when the ball will reach its maximum height.
(Round your answer to two decimal places.)
Step 2. Determine what the maximum height will be.
(Round your answer to two decimal places.)
--------------------------------------------------------------------Function notation and Graphing Functions
3
40. Given (𝑥) = √−4𝑥 + 4 , find 𝑔(−11) . Your answer should be simplified and written in
radical notation.
41.
For the function 𝑓(𝑥) = √𝑥 − 2
a. Find the domain.
b. Graph the function.
c. Find the range.
3
42.
Answer these questions about this quadratic function.
a. Does the parabola open up or down?
b. Determine the following points:
(1) Vertex
(2) x-intercepts(s), if any
(3) y-intercept
c. What is the equation for the line of symmetry?
d. Graph the function using at least 4 points.
e. What is the domain?
f. What is the range?
𝑦 = −𝑥 2 − 8𝑥 − 17
--------------------------------------------------------------------------------------------------Linear Inequalities
Solve these inequalities. Show your solution on number line and in interval notation.
43. |2𝑦 − 7| − 3 < 0
44. 2|𝑥 − 1| + 3 ≥ 5
45.
Solve the system of two linear inequalities graphically.
4
6𝑥 + 6𝑦 < 36
{
𝑥 ≥ 2