Math 154B
Elementary Algebra
Spring 2012
Final Exam Study Guide
The exam is on Wednesday, May 30th from 7:00pm–9:50pm. You are allowed a scientific calculator and a
4" by 6" index card for notes. On your index card be sure to write any formulas you needed for any of the
problems listed. I will not provide you with any formulas on the exam. The Final Exam is comprehensive;
however, not all of the problems that appeared in the previous exams will appear in the final. Use this
study guide to know which math concepts you need to review.
For the Final Exam, you will need to be able to:
1. Simplify expressions using the rules of exponents, including expressions with both positive, negative
and zero exponents. The final answer must have only positive exponents. 6.1 & 6.2
2. Convert to and from scientific notation and perform calculations in scientific notation. 6.3
3. Add or subtract two polynomials. Write the answer in descending order. 6.4
4. Multiply two polynomials. Write the answer in descending order. 6.5
5. Divide a polynomial by a monomial or a binomial. If there is a remainder, use appropriate notation.6.6
6. Factor completely any given polynomial using the methods learned. 7.1–7.5
a) Factor a polynomial by factoring out the GCF of all the terms. 7.1
b) Factor a polynomial by grouping. 7.2
c) Factor a trinomial with a leading coefficient that is one. 7.3
d) Factor a trinomial with a leading coefficient that is not one. 7.4
e) Factor a difference of squares. Recognize that a sum of squares is prime. 7.5
f) Factor using a mixture of these tools. 7.5
7. Solve a quadratic equation by factoring. *7.6
8. Simplify a rational expression and state the values for which the expression is defined. 8.1
9. Multiply or divide two rational expressions. Simplify the result. 8.2
10. Add or subtract two rational expressions that have common denominators or different denominators.
Simplify the result, if possible. 8.3 & 8.4
11. Solve a rational equation. Remember to eliminate solutions that make the expression in the original
equation undefined (make one of the denominators = 0). 8.6
12. Solve an application problem involving work or involving motion (time=distance/rate) 8.7
13. Simplify a square root expression by using the multiplication property of square roots or using a factor
tree and dividing variable exponents by 2. You might first need to multiply two square root
expressions and then simplify. 9.2
14. Add or subtract square root expressions. You might need to simplify terms before they can be
combined. 9.3
15. Multiply square root expressions by using the distributive property or the “FOIL” method. 9.3
16. Simplify a square root expression by using the division property of square roots. You might first need
to divide two square root expressions and then simplify. 9.4
17. Simply expressions with a square root in the denominator by rationalizing the denominator. You
might need to simplify first. 9.4
18. Solve a radical equation. If the equation has no solution, state so. Remember to check for extraneous
solutions. (In the process of solving the radical equation you might need to solve a quadratic
equation.) 9.5
19. Solve application problems involving square roots—the only problems I will put on the exam will
involve the Pythagorean Theorem. 9.6
20. Solve a quadratic equation by using the square root property. If possible, simplify radicals and
rationalize denominators. No decimal approximations. *10.1
21. Solve a quadratic equation by completing the square. If possible, simplify radicals and rationalize
denominators. No decimal approximations. *10.2
22. Solve a quadratic equation by using the quadratic formula. If possible, simplify radicals and
rationalize denominators. No decimal approximations. *10.3
*You must solve the quadratic equations using the indicated method in order to receive full credit.*
23. Solve application problems involving a quadratic equation. You will either be given the quadratic
equation and you will need to know how to use it to answer the question OR you
will need to come up the equation yourself—these involve the area of a rectangle (page 456 #20) or a
right triangle so you will use the Pythagorean Theorem (page 456 #35). You can solve the equation by
either factoring or using the quadratic formula (if the equation doesn’t factor) 7.7, 10.2, and 10.3
.
Repeating from above! On your index card be sure to write any formulas you needed for any of the
problems listed above. I will not provide you with any formulas on the exam.
Practice Problems for the Final
To study for the final do the following problems AND look at the problems that were on the exams (but
only those similar to those listed in this handout). The answer to the problems listed below, even and odd,
are in the back of the book. (For those of you who have the Chapter Test Prep Video cd that came with
the book, you can view it to see someone working out each of the problems that are in the Chapter Tests.)
Review Exercises Chapter 6
Page 398 #28, 32, 45, 108, 118, 123, 134
Review Exercises Chapter 7
Page 461 #82, 83, 85, 88, 101, 102, 114 (solve by factoring and applying the Zero-Product Rule), 127,
130
Review Exercises Chapter 8
Page 532 #12, 28, 58, 59, 60, 80, 81, 82, 85
Review Exercises Chapter 9
Page 588 #11, 18, 26, 32, 42, 47, 70
Also try:
Page 555 # 58
Page 570 #26 [answer: x=7], 31, 47
Chapter 10 Practice Test
Page 632 #2*, 6*
Also try page 630 #*7, *22, *37
*You must solve the quadratic equations using the method indicated. If you do not, then you will receive
minimal credit for a correct answer.
Math 154B
Final Exam Review
Name____________________
1. Simplify.
(2m 3 p 3 n)3
( mp 2 n 2 ) 3
2. Evaluate.
3. Evaluate:
1
2 4
4. Simplify:
( 4 x 3 y 2 z 4 )3 ( yz 4 )5
5. Convert the following to standard form.
3.79 10 5
6. Convert the following to scientific notation.
562000000000
7. Divide. Leave answer in scientific
notation.
(5 10 2 )
( 25 10 5 )
8. Multiply. Leave answer in scientific
notation.
9. Subtract.
(6 x 2 3 x 7 ) ( 4 x 3 3 x 2 5 x )
10. Multiply.
(2 x 1)(5 x 2
11. Multiply.
1
1
(n
)(n
)
5
5
12. Multiply.
x(x 2) 2
( 32 )3
(9 10 7 )(8 10 2 )
x 1)
13. Divide.
49 x 4 y 3 63 xy 2
7x2 y
3
28 x y
14. Divide using long division.
( x 3 2 x 2 3) ( x 1)
15. Factor
1 4 4 3
x
x
5
5
16. Factor
17. Factor
4x 2 9
18. Factor
25 x 2 25 xy
19. Solve for w by factoring.
w(w 5)
6
20. Solve for p by factoring.
3 p2 8 p 3 0
x2
5 x 36
6 y2
21. Multiply:
w 2 3w 2 w 1
w 2 4w 3 w 1
22. Divide:
2
2k 6
3 k k2 9
23. Add:
3
2
a 9
24. Subtract:
2p
4
2
p 1 p 1
a
2
2
6a 9
25. Solve for y .
3
4
6
2
y 2 y y 2y
26. Solve for a .
a
2a 16
2a 2 4a 4
16
a 1
27. Together, you and your brother can paint a room in 3 hours. Alone, your brother can paint a room
in 6 hours. How long would it take you to paint a room alone?
28. The wind is blowing at an average of 10 miles per hour. Riding with the wind, a bicyclist can
cycle 75 miles in the same amount of time it takes to cycle 15 miles against the wind. What is the
cyclist’s average rate in calm air?
Let
=
Rate
Distance
dist
rate
time
With the wind
Against the wind
29. Find each indicated root. If the root is not a real number, say so.
20
400
a.
b.
30. Find the missing length.
x
10
8
c.
121
d.
Simplify each expression completely. Assume all variables are real, non-negative numbers.
31. Multiply:
6
32. Multiply:
2
3
21 14
12
33. Multiply:
2x2 y
34. Rationalize the denominator:
6 xy 2
35. Simplify:
2 3 4
48a b c
37. Simplify:
1
1
27
75
3
5
2 14
2
36. Simplify:
3 28 4 7 2 63
38. Add:
4k 2 12n 2k 3k 2 n
39. Simplify:
(3 2 2) 2
40. Rationalize the denominator:
41. Rationalize the denominator:
42. Solve for x :
4x 1 1 x
2
1
4 2
32x
2
43. Simplify.
4 1
a 2a
a
a
2
44. Solve by using the square root property:
(3k 4) 2 25
45. Solve by completing the square:
x 2 5 2x
46. Solve by using the quadratic equation:
3x 2 x 2 0
47. Find the length and width of a rectangle if the width is 3 inches less than the length and the area of
the rectangle is 180 square inches.
Let ___________ = _____________________________________
Then ___________ = _____________________________________
Equation:
48. A 13-ft ladder is leaning against a building. The ladder reaches up the wall 7 feet more than the
distance of the ground between the bottom of the ladder and the base of the building. Find the
distance of the ground between the bottom of the ladder and the base of the building, and find the
distance the ladder reaches up the wall.
Let ___________ = _____________________________________
Then ___________ = _____________________________________
Equation:
__________ ___________

 
 

2
(leg)
2
(other leg)
__________

 

(hy potenuse)2
49. A rectangle has a diagonal of 17 feet. If the length of the rectangle is 1 foot less than twice its
width, find the dimensions of the rectangle.
Let ___________ = _____________________________________
Then ___________ = _____________________________________
Equation:
50. The equation for the height of a ball thrown, from a 160 foot cliff, into the air at 64 feet per second
16 t 2 64 t 160 , where h(t ) is the height of the ball after t seconds.
is h(t )
a.) Calculate the time it takes for the ball to be 208 feet above the ground.
Equation: _______



208 ft ------------------------160 ft
t
?
t
_____________________

height
equation
?
b.) Calculate the seconds it takes for the ball to hit the ground.
Equation: _______



height
160 ft
0 ft --
t
?
_____________________

equation
Answers:
1.
8m 6 n 9
p15
9 11 32
64 x y z
4.
7. 2 106
10. 10x3 7 x 2 x 1
9y
4x
x
16. ( x 9)(x 4)
19. w 2, w 3
13. 7 x 2 y 2
22. -1
25. No solution
28. x 15 mph
2. -729
3. 16
5. 0.0000379
6. 5.62 1011
8. 7.2 106
1
11. n 2
25
14. x 2 3x 3
9.
12.
17. (2 x 3)(2 x 3)
1
20. p 3, p
3
5a 3
23.
(a 3) 2 (a 3)
26. a 20
2 5 , not real
31. 6 2
29. 20,
1
32.
7
34. 2 7
2
35. 4abc 3b
37. 2 3
x
x
3
43. 2
a
40.
2
3
49. Width = 8 feet
Length = 15 feet
46. x 1, x
2( p 2 )
p 1
27. x 6 hours
30. x 6
24.
36. 8 7
39. 22 12 2
38. 10 k 3n
41. 2 2
42. x
1
3
47. Width = 12 inches
Length = 15 inches
50. a. 1 and 3 seconds
b. 2
1 3
x ( x 4)
5
18. (5 x 3 y)(5 x 2 y)
w 2
21.
w 3
15.
33. 2xy 3xy
2
44. k
4 x 3 3x 2 8 x 7
x3 4 x 2 4 x
3, k
14
5.74 seconds
45. x
6
1
6
48. 5 feet and
12 feet up the wall