Name: ________________________ Class: ___________________ Date: __________
Test 2 Review (Math1650, $3.3-3.7 & Chap.4)
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Divide P(x) by D(x) and express P(x) in the form P (x )  D (x )  Q (x )  R (x ) .
P(x)  x 3  3x 2  5x  1, D(x)  x  1
Ê
ˆ
a. P(x)  (x  1)  ÁÁ x 2  4x  1 ˜˜  0
Ë
¯
ˆ
ÊÁ 2
b. P(x)  (x  3)  Á 2x  4x  1 ˜˜  3
¯
Ë
ˆ
Ê 2
Á
c. P(x)  (x  1)  Á 2x  4x  2 ˜˜  1
¯
Ë
ˆ
ÊÁ 2
d. P(x)  (x  2)  Á 2x  3x  2 ˜˜  1
¯
Ë
ˆ˜
ÊÁ 2
e. P(x)  (x  3)  Á x  6x  2 ˜  3
¯
Ë
____
2. Divide P (x ) by D (x ) and express P (x ) in the form P (x ) = D (x ) Q (x ) + R (x ) .
P (x ) = x 3 + 2x 2  4x + 1, D (x ) = x  1
a.
b.
c.
d.
e.
____
Ê
ˆ
P (x )  (x  3 )  ÁÁÁ 2x 2  3x  1 ˜˜˜  3
Ë
¯
Ê 2
ˆ
Á
˜
P (x )  (x  3 )  ÁÁ x  x  2 ˜˜  3
Ë
¯
ÊÁ 2
ˆ
P (x )  (x  1 )  ÁÁ x  3x  1 ˜˜˜  0
Ë
¯
ÊÁ 2
ˆ
P (x )  (x  1 )  ÁÁ 2x  3x  2 ˜˜˜  1
Ë
¯
ÊÁ 2
ˆ˜
P (x )  (x  2 )  ÁÁ 2x  2x  2 ˜˜  1
Ë
¯
3. Find the quotient and remainder using long division.
4x 3 + 6x 2 + 6x
2x 2 + 1
a.
b.
c.
d.
e.
The quotient is 2x  3; the remainder is 4x  2.
The quotient is 4x  3; the remainder is 2x  3.
The quotient is 2x + 3; the remainder is 4x  3.
The quotient is 4x  3; the remainder is 2x + 3.
no solution given
1
ID: A
Name: ________________________
____
ID: A
4. Find the quotient and remainder using long division.
6x 2  7x + 5
2x 2  3x
a.
b.
c.
d.
e.
____
The quotient is 2x + 5; the remainder is 3.
The quotient is 2x  5; the remainder is 3.
The quotient is 3; the remainder is 2x  5 .
The quotient is 3; the remainder is 2x + 5 .
no solution given
5. Find the quotient and remainder using synthetic division.
x 4  5x 3 + 7x 2  228x  156
x8
____
a.
The quotient is x 3  3x 2  31x  20 ; the remainder is 4.
b.
The quotient is x 3  3x 2  31x  20 ; the remainder is 4.
c.
The quotient is x 3  3x 2  31x  20 ; the remainder is 4.
d.
The quotient is x 3  3x 2  31x  20 ; the remainder is 4.
e.
The quotient is x 3  3x 2  31x  20 ; the remainder is 4.
6. Find a polynomial of degree 3 that has zeros 8,8,and 4.
a.
b.
c.
d.
e.
____
x 3  4x 2  64x  256
x 3  4x 2  64x  256
x 3  4x 2  64x  256
x 3  4x 2  64x  256
x 3  4x 2  64x  256
7. List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are
zeros).
U(x)  6x 5  6x 3  2x  12
a. 1,  2,  3,  4,  6,  12
1
3
1
2
4
1
b. –1, –2, –3, –4, –6, –12,  ,  ,  ,  ,  , 
2
2
3
3
3
6
1
3
1
2
4
1
c. 1,  2,  3,  4,  6,  12,  ,  ,  ,  ,  , 
2
2
3
3
3
6
1 3 1 2 4 1
d. 1, 2, 3, 4, 6, 12, , , , , ,
2 2 3 3 3 6
e. –1, –2, –3, –4, –6, –12
2
Name: ________________________
____
ID: A
8. Find all rational zeros of the polynomial:
P(x) = x 3 + 3x 2  4
a.
b.
c.
d.
e.
____
x = 1, x = 2
x =  1, x =  2
x = 1, x =  2
x = 1, x =  1
1
1
x = ,x =
1
2
9. Find all rational zeros of the polynomial.
P (x) = x 4  20x 2 + 64
a.
b.
c.
d.
e.
x =  17, x = 17, x =  2, x = 2
x =  4, x = 4, x = 2
x =  3, x = 3, x =  2, x = 2
x = 4, x =  2, x = 2
x =  4, x = 4, x =  2, x = 2
____ 10. Find integers that are upper and lower bounds for the real zeros of the polynomial.
P (x) = x 3  24x 2 + 126x + 16
a.
b.
c.
d.
e.
x   24, x   1
x   1, x   24
x  1, x  24
x  0, x  1
x   1, x  24
____ 11. Find the real and imaginary parts of the complex number.
8
a.
b.
c.
Real part 0, imaginary part 8
Real part 8, imaginary part 0
Real part 8, imaginary part 8
____ 12. Evaluate the expression (9 + 14i) + (7 – 11i) and write the result in the form a + bi.
a. 16 + 3i
b. 16 – 3i
c. 9 + 14i
d. 3 + 16i
____ 13. Evaluate the expression (4 + 9i)(11 – 10i) and write the result in the form a + bi.
a. 44 + 99i
b. –59 – 134i
c. 134 + 59i
d. 59 + 134i
3
Name: ________________________
ID: A
____ 14. Evaluate the expression i 17 and write the result in the form a + bi.
a. –( i )
b. i
c. –1
d. 1
____ 15. Evaluate the expression i 64 and write the result in the form a + bi.
a. –( 1 )
b. i
c. 1
d. –i
____ 16. Evaluate the expression
a.
b.
c.
d.
35
40

and write the result in the form a + bi.
2i 2i
2 + 15i
–15 – 2i
2 – 15i
15 – 2i
____ 17. Find the polynomial P (x ) of degree 3 with integer coefficients, and zeros 3 and 2i .
a.
b.
c.
d.
e.
3x 2  5x  6
x 3  3x 2  4x  3
x 3  5x 2  3x  12
x 3  5x 2  6x
x 3  3x 2  4x  12
____ 18. Find the x- and y-intercepts of the rational function r ( x ) =
a.
b.
c.
d.
e.
x-intercept (–18, 0), y-intercept (0, –2)
x-intercept (–3, 0), y-intercept (0, 18)
x-intercept (18, 0), y-intercept (0, –3)
x-intercept (18, 0), y-intercept (0, –5)
x-intercept (–1, 0), y-intercept (0, 18)
____ 19. Find the vertical asymptote of the rational function r ( x ) =
a.
b.
c.
d.
e.
x  18
.
x+6
x = 9
x = 1
x = 18
x=1
x=9
4
x2 + 1
.
x9
Name: ________________________
ID: A
____ 20. Find the intercepts and asymptotes of the rational function r ( x ) =
9x + 108
.
4x + 12
a.
x-int.
(–12, 0)
y-int.
(0, 9)
horiz. asymptote
vert. asymptote
x = –2.25
b.
x-int.
(–12, 0)
y-int.
(0, 9)
horiz. asymptote
vert. asymptote
x = –9
c.
x-int.
(0, –12)
y-int.
(9, 0)
horiz. asymptote
vert. asymptote
x = 3
d.
x-int.
(–12, 0)
y-int.
(0, 9)
horiz. asymptote
vert. asymptote
x = 3
e.
x-int.
(–12, 0)
y-int.
(0, 9)
horiz. asymptote
vert. asymptote
x = –2.25
y4
y9
y  2.25
y  2.25
y3
____ 21. Find the y-intercept and asymptotes of the rational function r ( x ) =
a.
b.
c.
d.
e.
y-intercept
(0, 3)
y-intercept
(0, 5)
y-intercept
(0, 3)
y-intercept
(0, 3)
y-intercept
(0, 5)
horizontal
y =
horizontal
y =
horizontal
y =
horizontal
y =
horizontal
y =
asymptote
5
asymptote
0
asymptote
0
asymptote
1
asymptote
1
____ 22. Find the slant asymptote of the function y =
a.
b.
c.
d.
e.
x2
.
x1
y=x+1
y=x3
y=x+5
y=x+4
y=x2
5
vertical
x
vertical
x
vertical
x
vertical
x
vertical
x
75
.
( x  5) 2
asymptote
= 0
asymptote
= 3
asymptote
= 5
asymptote
= 5
asymptote
= 3
Name: ________________________
ID: A
____ 23. Find the exponential function f(x)  a x whose graph is given.
a.
b.
c.
d.
e.
f(x)  4 x
f(x)  4 x  4
f(x)  4 x
f(x)  x 4
f(x)  4 x
____ 24. If $1,000 is invested at an interest rate of 10% per year, compounded semiannually, find the value of the
investment after 10 years.
a. $1629
b. $2653
c. $1000
d. $2753
e. $377
____ 25. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to
produce the desired sum at a later date. Find the present value of $1,000 if interest is paid at a rate of 6% per
year, compounded semiannually, for 8 years.
a. $627
b. $623
c. $875
d. $1605
e. $1594
6
Name: ________________________
ID: A
____ 26. Graph the function, not by plotting points, but by starting from the graph in the figure. State the domain,
range, and asymptote.
y  ex  3  4
7
Name: ________________________
a.
ID: A
d.
Domain: (4,). Range: (,).
Asymptote: y  3.
Domain: (3,). Range: (4,).
Asymptote: y  3.
e.
b.
Domain: (3,). Range: (,).
Asymptote: x  3.
Domain: (,). Range: (4,).
Asymptote: y  4.
8
Name: ________________________
ID: A
c.
Domain: (3,). Range: (,).
Asymptote: y  4.
____ 27. A radioactive substance decays in such a way that the amount of mass remaining after t days is given by
m(t)  12e 0.011t
where m(t) is measured in kilograms. How much of the mass remains after 25 days?
a. 9.02 kg
b. 15.80 kg
c. 12.76 kg
d. 9.22 kg
e. 9.11 kg
____ 28. The population of a certain species of bird is limited by the type of habitat required for nesting. The
population behaves according to the logistic growth model
n(t) 
500
0.2  21.7e 0.385t
where t is measured in years. What size does the population approach as time goes on?
a. 2500
b. 7500
c. 5000
d. 500
e. 100
9
Name: ________________________
ID: A
____ 29. Express the equation in exponential form.
log 4 16  2
a.
b.
c.
d.
e.
2 4  16
4 2  16
none of these
2 16  4
16 2  4
____ 30. Express the equation ln (x + 1) = 4 in exponential form.
a. none of these
b. x  e 1  4
c. x  e 4  1
d. x  e 1  4
e. x  e 4  1
____ 31. Express the equation in logarithmic form.
10 3 = 1,000
a.
b.
c.
d.
e.
log 3 10 = 1,000
log 3 1,000 = 10
log 10 1,000 = 3
log 1,000 10 = 3
none of these
____ 32. Express the equation in logarithmic form.
e x  2  0.2
a. x = 0.2 – ln 2
b. x = 0.2 + ln 2
c. x = 2 + ln 0.2
d. x = –2 + ln 0.2
e. none of these
____ 33. Evaluate the expression.
e ln 5
a.
b.
c.
d.
e.
5e
5
none of these
ln5
e5
10
Name: ________________________
ID: A
____ 34. Evaluate the expression.
10
a.
b.
c.
d.
e.
log 

log 
none of these
 10
1
____ 35. Use the definition of the logarithmic function to find x.
log x 81  4
a. x = 81
b. none of these
c. x = 5
d. x = 4
e. x = 3
____ 36. Find the function of the form y  log a x whose graph is given.
a.
b.
c.
d.
e.
y  log 8 (x)
none of these
y  log 2 (x)
y  log 3 (x)
y  log 5 (x)
11
Name: ________________________
ID: A
____ 37. Use the graph of y = log 3 x below to help you identify the graph of y = 3 x .
a.
d.
b.
e.
c.
12
none of these
Name: ________________________
ID: A
____ 38. Find the domain of the function.
f(x)  x  5  log 3 (11  x)
a. [–11, –5)
b. none of these
c. [5, 11]
d. [–5, 11]
e. [5, 11)
____ 39. Find the domain of the function.
ˆ
Ê
f(x)  log 7 ÁÁ x  x 10 ˜˜
¯
Ë
a. (–1, 1)
b. none of these
c. (1,)
d. (0,)
e. (0, 1)
____ 40. Evaluate the expression.
log 3 189 – log 3 7
a. 21
b. ln 189
c. 7
d. 3
e. log 3 182
____ 41. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product,
quotient, or power.
log 7
a.
b.
c.
d.
e.
x2  5
ˆ
Ê
8log 7 ÁÁ x 2  5 ˜˜
¯
Ë
x2  5
log 7
8
1Ê
Á 2log 7 s  log 7 5 ˆ˜
¯
8Ë
Ê
ˆ
log 7 ÁÁ x 2  5 ˜˜
Ë
¯
1
ÊÁ 2
ˆ˜
log Á x  5 ˜
8 7Ë
¯
8
13
Name: ________________________
ID: A
____ 42. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product,
quotient, or power.
ln
a.
b.
c.
d.
e.
9
3r 8 s
8
8
8
ln3  lnr  lns
9
9
9
8
1
1
ln3  lnr  lns
9
9
9
8
1
ln3  lnr  ln s
9
9
1
1
1
ln3  lnr  lns
9
9
9
8
ln3  ln r  ln s
9
____ 43. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product,
quotient, or power.
ÁÊÁ a 6
log ÁÁÁÁ 3
Áb
c
Ë
a.
b.
c.
d.
e.
˜ˆ˜
˜˜
˜˜
˜
¯
1
log c
2
1
6 loga  3log b  log c
2
c
6 loga  3 logb  log
2
1
6 loga  log b  log c
2
1
log(6a)  3log b  logc
2
6 loga  3log b 
14
Name: ________________________
ID: A
____ 44. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product,
quotient, or power.
ÊÁ
ˆ
ÁÁ
y ˜˜˜˜
Á
9
˜
ln ÁÁÁ x
z ˜˜˜˜
ÁÁ
Ë
¯
1
1
a. lnx  ln y  ln z
9
9
1
1
b. lnx  ln y  ln z
9
9
1
1
c. lnx  ln y  ln z
9
9
1
d.
(ln x  lny  ln z)
9
1
1
e. lnx  ln y  ln z
9
9
____ 45. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product,
quotient, or power.
log 6 x 6 y 6 z
a.
b.
c.
d.
e.
1
1
1
log x 
log y  logz
36
6
216
1Ê
Á logx  log y  log z ˜ˆ
¯
6Ë
1
1
1
logx 
log y 
logz
36
216
6
1 Ê
Á log x  log y  log z ˆ˜
¯
216 Ë
1
1
1
log x 
log y  logz
36
6
216
____ 46. Rewrite the expression below as a single logarithm.
d.
1
log3  log2
2
1
log
3
7
ln3 7
1
log
7
3
log 21
e.
log7 3
log14 
a.
b.
c.
15
Name: ________________________
ID: A
____ 47. Find the solution of the exponential equation.
e 4 2 x = 14
a.
b.
c.
d.
e.
x = 1.8105
x = 0.6805
x = –0.0963
x = 2.9391
x = 2.7183
____ 48. Find the solution of the exponential equation, correct to four decimal places.
12 x
a.
b.
c.
d.
e.
 5x  4
x = 7.3547
none of these
x = 1.544
x = 7.3535
x = 0.544
____ 49. Find the solution of the exponential equation, correct to four decimal places.
1.00808 5x  8
a. x = 0.625
b. x = 1.6
c. x = –1.6032
d. x = –51.679
e. x = 51.679
____ 50. Solve the equation.
e 2x  5e x  4  0
a. x = 4, x = 1
b. x = 1.3863, x = 0
c. x = 1.6094
d. x = 0.7213, x = 0
e. x = –4, x = 1
____ 51. Solve the equation.
e 2 x  8e x + 7 = 0
a.
b.
c.
d.
e.
x = 2.0794
x = –7, x = 1
x = 0.5139, x = 0
x = 7, x = 1
x = 1.9459, x = 0
16
Name: ________________________
ID: A
____ 52. Solve the logarithmic equation for x.
log 3 (4 – x) = 7
a. x = –2183
b. x = 2191
c. x = 2187
d. x = –2191
e. x = –2187
____ 53. Solve the logarithmic equation for x.
log 2 2 + log 2 x = log 2 3 + log 2 (x – 5)
a. x = 12
b. x = 30
c. x = 15
d. x = 17
e. x = 3.9
____ 54. For what value of x is the following true?
log (x + 9) = log x + log 9
a. x = –7.875
b. x = 0.051
c. x = 4.5
d. x = 1.125
e. x = 10
____ 55. Solve the inequality.
log (x – 2) + log (9 – x) < 1
a. x  (2, 9)
b. x  (4, 7)
c. x  (, 4)  (7, )
d. x  (, 2)  (9, )
e. x  (2, 4)  (7, 9)
____ 56. Solve the inequality.
x 2 e x  16e x  0
a.
b.
c.
d.
e.
x  (4, 0)
x  (0, 4)
x  (4, 4)
x  (4, 16)
x  (16, 16)
17
Name: ________________________
ID: A
____ 57. Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year,
compounded quarterly.
a. 13 years
b. none of these
c. 12 years
d. 50 years
e. 3 years
____ 58. How long will it take for an investment of $1,000 to double in value if the interest rate is 7.5% per year,
compounded continuously?
a. none of these
b. 14.65 years
c. 0.09 year
d. 9.24 years
e. 14.39 years
____ 59. A 13-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by
m(t)  13e 0.089t
where m( t ) is measured in grams. After how many days is there only 10 g remaining?
a. 2 days
b. 6 days
c. 3 days
d. 5 days
e. 4 days
____ 60. The population of California was 10,290,518 in 1940 and 23,626,378 in 1985. Assume the population grows
exponentially. Find the time required for the population to double (in years).
a. 37.53 yr
b. 54.14 yr
c. 41.23 yr
d. 108.28 yr
e. 0.83 yr
____ 61. The half-life of cesium-137 is 30 years. Suppose we have a 17-g sample. Find a function that models the mass
remaining after t years.
a. m ( t ) = 20e - 0.03t
b. m ( t ) = 20e - 0.02t
c. m ( t ) = 17e - 0.024t
d. m ( t ) = 17e - 0.023t
e. m ( t ) = 30e - 0.023t
____ 62. Radium-221 has a half-life of 30 s. How long will it take for 95% of a sample to decay?
a. 2.22 s
b. 44.94 s
c. 1.54 s
d. 129.66 s
e. 89.87 s
18
Name: ________________________
ID: A
____ 63. An unknown substance has a hydrogen ion concentration of
ÈÍ  ˘˙
ÍÍ H ˙˙  6.1  10 3 M .
ÍÎ ˙˚
Find the pH.
a. pH = 2.2
b. pH = 12.9
c. pH = 3.0
d. pH = 5.1
e. none of these
19
ID: A
Test 2 Review (Math1650, $3.3-3.7 & Chap.4)
Answer Section
MULTIPLE CHOICE
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