Answers and Explanations

Answers and Explanations

1. The correct answer is (A). You can use FOIL 7. The correct answer is (D). Using the laws

to multiply (3x + 5)(3x + 5). Start by multiplying the first terms: (3x)(3x) = 9x2. Then multiply the outer terms: (3x)(5) = 15x and the inner terms: (5)(3x) = 15x. Finally, multiply the last terms: (5)(5) = 25. Add the terms and

of exponents,

you

can

see that

4 = 4 ; 23

8

( ) ( ) ( ) 3

42 = 46;

24 3 = 212; 2 43 = 264 ; and

2

34 = 38.

Clearly, choice D will result in the greatest number.

you get 9x2 + 30x + 25.

8. The correct answer is (E). Rewrite the factori-

2. The correct answer is (C). 16 ? (3x + 2)2 is the

als in the fraction to simplify this expression:

difference of two squares. Remember that a2 ? b2 = (a + b)(a ? b). So, in this case, 16 ? (3x + 2)2 = (4 + (3x + 2))(4 ? (3x + 2)).

(x

x!

- 1)!

=

(x)(x -1)( (x -1)(x

x - 2)(x - 2)(x -

- 3)...(1) 3)...(1)

.

3. The correct answer is (C). To solve this prob-

After simplifying the fraction, the expression is

lem, you rewrite the equation with the correct

x + x, or 2x.

signs by distributing the negative sign: 9. The correct answer is (B). Remember the rule

4r(2r2 ? 6) ? 5r3 + 12 ? 10r + 8. Multiply 4r by

(2r2 ? 6), and then combine like terms.

of negative exponents: a-n = 1 and of products

an

4. The correct answer is (E). By rewriting a + b =

with the same base: am ? an = am+n. Use these rules

4, you can see that |4 ? a| + |4 ? b| is the same as

to help you answer the question.

|b| + |a|. Since you do not know whether the values for a and b are positive or negative, you cannot determine the sum of their absolute values.

10. The correct answer is (D). You need to find the total number of feet between the first and eighth bounce, so i = 1 and the upper limit is 8. The ball

5. The correct answer is (B). Remember that di-

bounces down and then back up, so to find the

viding by a fraction is the same as multiplying

total number of feet, you need to multiply by 2h.

by the fraction's inverse.

11. The correct answer is (A). The graph of y = ?5

x2 - 16

x+5 x+4

=

x2 x

- 16 +5

x x

- +

5 4

will be a horizontal line that passes through (0, ?5).

12. The correct answer is (E). Remember that when

x-5

Once you factor x2 ? 16, you will be able to find the correct answer.

you work with absolute value and inequalities, you need to find the solution to two inequalities. To solve this problem, you need to find the solutions to p + 8 12 and p + 8 ?12.

6. The correct answer is (A). Try listing the fac- 13. The correct answer is (E). You will need to use

tors of ?28. Then list the factors of 8x2. Which

the Quadratic Formula to answer this question:

combination of factors results in the expression

above? You might also solve this problem by first factoring the greatest common factor, 2.

x = -b ? b2 - 4ac . First rewrite the equation 2a

in the form ax2 + bx + c = 0. Be sure to plug in

the correct values for a, b, and c and solve for x.

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CLEP College Algebra

14. The correct answer is (A). You can pick a point that is in the shaded region (like (1, 2)) and plug in those values for x and y in each equation. Those values will make the correct equation true. If the point you choose satisfies more than one equation, pick another point in the shaded region.

15. The correct answer is (C). First rewrite the equa-

tion in standard form: y = a x - 1 . Recall that a

bb

b

represents the slope of the line. Since the line goes

up from left to right, its slope is positive. Find the

answer choice that will yield a positive slope.

16. The correct answer is (D). You need to use the discriminant to solve this problem. When b2 ? 4ac 0, there will be one or two real roots.

17. The correct answer is (E). Since the boundary line is y = ?5x + 6, you can see that the y-intercept is (0, 6), and that the line has a negative slope. Make a table of xy values to determine the quadrants in the graph of 5x + y < 6.

18. The correct answer is (C). The value of Janna's account is the original amount, plus $250 per year. The value of Emily's car is the original amount minus $500 per year. How can you show "per year" in math?

19. The correct answer is (D). Remember that A B ={45,50,60,65,75}. Which numbers does this set have in common with C?

20. The correct answer is (A). i is equal to the square

root of ?1, which not a real number. 3.33 is rational because repeating decimals are rational. Although is irrational, the approximation repre-

sented by 22 is rational because it is a fraction 7

made up of integers.

21. The correct answer is (2). To solve this prob-

lem, you need to multiply 2 - 6i by 2 + i , since 2-i 2+i

2 +i is the complex conjugate of the denominator. When you combine like terms after the multipli-

cation, remember that i = -1, so i2 = ?1.

22. The correct answer is (D). Assign values for l, m, and n and determine which inequality does not make sense. For example, if l = 3, m = 2, and n = ?5, which answer choice is not true?

23. The correct answer is (B). Try writing the equa-

( )( )( )( )( ) tion as 3i 3i 3i 3i 3i . For each ( )( ) 3 3 , the product is 3, and for each (i)(i),

the product is ?1.

24. The correct answer is (C). To solve this problem, substitute each value for y into

5 = 1 + y . Find the value or values of y y + 7 5 10 that make the equation true.

25. The correct answer is (B). Use the formula for finding the nth term in an arithmetic sequence: nth term = a + (n ? 1)(d), where a = the first term, n = the total number of terms, and d = the common difference.

26. The correct answer is (C). To find the answer, use the table to determine the number of times that the value of p(r) changes signs.

27. The correct answer is (324). This question is about an arithmetic sequence: 35, 33, 31, ...Once you have determined the total number of terms in the sequence, you can use the formula for finding the sum of the

[ ( ) ] first n terms: S = n 2a + n -1 d , where n = the 2 total number of terms, a = the first term, and d = the

common difference between terms.

28. The correct answer is (D). To solve this problem,

you need to use the Binomial Formula for

n(n -1)

(a + b)n : (a + b)n = an + nan?1b +

an?2b2 +

2!

( )( ) n n -1 n - 2 an?3b3 + ... + bn. In this case, a = x ,

3!

3

b = 2x, and n = 6. To find the coefficient, you can

also use Pascal's Triangle. The exponents of each

variable will be half of the exponent (6) of the bi-

nomial, since you are looking for the middle term.

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CLEP College Algebra

29. The correct answer is (a = 10). Substitute 5 for x in the equation and solve for a: 2(53) ? a(52) + 4(5) ? 20 = 0.

30. The correct answer is (A). You will need to remember the formula for finding the nth term of a geometric sequence: nth term = arn?1, where a = the first term and r = the common ratio. You could also test the given terms in the answer choice. For example, if n = 1, which answer

choice equals 1 ? 5

31. The correct answer is (D). When you use synthetic division to solve this problem, you should end up with a remainder of 3l + 1. Since you know the remainder is 19, then 3l + 1 must be equal to 19. Then solve for l.

32. The correct answer is C. First find f(0): What is the value of y when x = 0? Substitute this value for x and find its corresponding y-value.

33. The correct answer is (25). To find the x-intercept, substitute 0 for y and solve for x: 0 = 3 x -15. 5

34. The correct answer is (D). The graph in I does not show a function because there is more than one y-value for each x-value. II and III are functions because there is exactly one y-value for every x-value.

35. The correct answer is (2). Rewrite 25p as (52)p. Then use the rules of exponents to solve for p.

36. The correct answer is (D). You should remember that the graph of an equation involving x2 will always result in a parabola that opens up or down.

37. The correct answer is (B). To find the answer, you need to substitute 3x + 3 for x in 4x ? 3: 4(3x + 3) ? 3. Now evaluate the expression.

38. The correct answer is (C). Remember that you can rewrite logs with exponents: log2(x ? 6) = 5 is the same as 25 = x ? 6. Now you can solve for x.

39. The correct answer is (B). You may notice that this problem involves an exponential function. This function is defined by 8(2x), where x = each period of 20 minutes (8 comes from the initial number of milliliters and 2 represents the doubling of the number of milliliters after every 20 minutes). Find the number of 20-minute periods, plug that value in for x, and solve.

40. The correct answer is (C). III is incorrect. If you rewrite the equation using exponents, you will see that it is not true: 44x 4x.

41. The correct answer is (E). To find the inverse function, switch the ys and xs in the original equation: x = 8 + 3y2. Then solve for y.

42. The correct answer is (B). Before you can subtract, you need to give the fractions common de-

nominators. Multiply 3x - 2 by 3x + 2 and x + 4 3x + 2

x - 5 by x + 4 . Now both fractions will have 3x + 2 x + 4 the denominator of (x + 4)(3x + 2) and you can subtract.

43. The correct answer is (A). This graph is the only one that shows a shift of up 2 and over 2.

44. The correct answer is (A). First use FOIL to find the product of (i ? 4)(2 + i). Then add (3i ? 6).

Remember that i 2= -1.

45. The correct answer is (m = 2). You know that mn4 = 162 and mn3 = 54. You can divide to find the value of n, and once you know the value of n, you can find the value of m.

46. The correct answer is (B). Remember the definition of a perfect square trinomial: (a + b)2 = a2 + 2ab + b2. To find the value of 2ab, multiply

9x2 by 16 by 2.

47. The correct answer is (C). To solve this problem, plug each set of values into the equations. The values that satisfy both equations will be the correct answer.

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CLEP College Algebra

48. The correct answer is (B). To find the correct point, count one unit to the left on the real axis, and then 4 units up on the imaginary axis.

49. The correct answer is ($7,891). You need

to know the formula for computing compound

interest in order to solve this problem:

nt

A

=

P1

+

r n

, where P = the principal amount,

r = the annual interest rate, n = the number of

times it is compounded per year, and t = the num-

ber of years.

50. The correct answer is (D). II is not true because the domain cannot be equal to 0, since 5y can never be equal to 0.

51. The correct answer is (D). Be sure to rewrite the equation with the correct signs before you combine like terms: 3x(4x2 + 2) ? 8 + 2x3 ? 10x ? 4.

52. The correct answer is (A). Remember that

AB=

1 4

,

2 5

,

3 5

,

4 6

,

7 8

.

Which

numbers

does

this set have in common with C?

53. The correct answer is (B). 6.8989... is rational

because it is a repeating decimal. 5 is irrational because it results in a nonrepeating, nonterminating decimal. is irrational for the same reason.

54. The correct answer is (B). You will need to remember the formula for finding the nth term of a geometric sequence: nth term = arn ? 1, where a = the first term and r = the common ratio.

55. The correct answer is (C). You can rewrite a + 5 = b as 5 = b ? a and a ? b = ?5. Use this information to determine the correct answer.

56. The correct answer is (E). To locate the point, count 3 units to the left and 2 units down.

57. The correct answer is (D). Use the formula for finding the nth term in an arithmetic sequence: nth term = a + (n ? 1)(d), where a = the first term, n = the total number of terms, and d = the common difference.

58. The correct answer is (B). To find the answer, use the table to determine the number of times that the value of f(x) changes signs.

59. The correct answer is (D). III is imaginary be-

( ) cause 2i = 2i = 2 -1 , and is not rational.

60. The correct answer is (A). Make sure you keep track of the positive and negative signs in this equation. You can rewrite the equation as ?5b(b2 + 2b) ? 4 ? 3b3 + 6b + 7.

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