Name: ________________________ Class: ___________________ Date: __________ ID: A

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Test 2 Review (Math1650, $3.3-3.7 & Chap.4)

Multiple ChoiceIdentify 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) x3 3x2 5x 1, D(x) x 1

a. P(x) (x 1) x2 4x 1ÊËÁÁÁ

ˆ¯˜̃̃ 0

b. P(x) (x 3) 2x2 4x 1ÊËÁÁÁ

ˆ¯˜̃̃ 3

c. P(x) (x 1) 2x2 4x 2ÊËÁÁÁ

ˆ¯˜̃̃ 1

d. P(x) (x 2) 2x2 3x 2ÊËÁÁÁ

ˆ¯˜̃̃ 1

e. P(x) (x 3) x2 6x 2ÊËÁÁÁ

ˆ¯˜̃̃ 3

____ 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. P x( ) x 3( ) 2x2 3x 1ÊËÁÁÁ

ˆ¯˜̃̃ 3

b. P x( ) x 3( ) x2 x 2ÊËÁÁÁ

ˆ¯˜̃̃ 3

c. P x( ) x 1( ) x2 3x 1ÊËÁÁÁ

ˆ¯˜̃̃ 0

d. P x( ) x 1( ) 2x2 3x 2ÊËÁÁÁ

ˆ¯˜̃̃ 1

e. P x( ) x 2( ) 2x2 2x 2ÊËÁÁÁ

ˆ¯˜̃̃ 1

____ 3. Find the quotient and remainder using long division.

4x 3 + 6x 2 + 6x

2x 2 + 1

a. The quotient is 2x3; the remainder is 4x2.b. The quotient is 4x3; the remainder is 2x3.c. The quotient is 2x +3; the remainder is 4x3.d. The quotient is 4x3; the remainder is 2x +3.e. no solution given

Name: ________________________ ID: A

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____ 4. Find the quotient and remainder using long division.

6x 2 7x + 5

2x 2 3x

a. The quotient is2x +5; the remainder is 3.b. The quotient is2x5; the remainder is 3.c. The quotient is 3; the remainder is 2x5.d. The quotient is 3; the remainder is 2x +5.e. no solution given

____ 5. Find the quotient and remainder using synthetic division.

x 4 5x 3 + 7x 2 228x 156x 8

a. The quotient is x 3 3x 2 31x20; the remainder is 4 .

b. The quotient is x 3 3x 2 31x20; the remainder is 4.

c. The quotient is x 3 3x 2 31x20; the remainder is 4 .

d. The quotient is x 3 3x 2 31x20; the remainder is 4.

e. The quotient is x 3 3x 2 31x20; the remainder is 4 .

____ 6. Find a polynomial of degree 3 that has zeros 8,8,and 4.

a. x 3 4x 2 64x256

b. x 3 4x 2 64x256

c. x 3 4x 2 64x256

d. x 3 4x 2 64x256

e. x 3 4x 2 64x256

____ 7. List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros).

U(x) 6x5 6x3 2x 12a. 1, 2, 3, 4, 6, 12

b. –1, –2, –3, –4, –6, –12, 12

, 32

, 13

, 23

, 43

, 16

c. 1, 2, 3, 4, 6, 12, 12

, 32

, 13

, 23

, 43

, 16

d. 1, 2, 3, 4, 6, 12, 12

,32

,13

,23

,43

,16

e. –1, –2, –3, –4, –6, –12

Name: ________________________ ID: A

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____ 8. Find all rational zeros of the polynomial:

P(x) = x 3 + 3x 2 4

a. x = 1, x = 2b. x = 1, x = 2c. x = 1, x = 2d. x = 1, x = 1

e. x =11

, x = 12

____ 9. Find all rational zeros of the polynomial.

P (x) = x 4 20x 2 + 64

a. x = 17, x = 17, x = 2, x = 2b. x = 4, x = 4, x = 2c. x = 3, x = 3, x = 2, x = 2d. x = 4, x = 2, x = 2e. 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. x 24, x 1b. x 1, x 24c. x 1, x 24d. x 0, x 1e. x 1, x 24

____ 11. Find the real and imaginary parts of the complex number.

8a. Real part 0, imaginary part 8b. Real part 8, imaginary part 0c. 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 + 3ib. 16 – 3ic. 9 + 14id. 3 + 16i

____ 13. Evaluate the expression (4 + 9i)(11 – 10i) and write the result in the form a + bi.a. 44 + 99ib. –59 – 134ic. 134 + 59id. 59 + 134i

Name: ________________________ ID: A

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____ 14. Evaluate the expression i 17 and write the result in the form a + bi.a. –( i )b. ic. –1d. 1

____ 15. Evaluate the expression i 64 and write the result in the form a + bi.a. –( 1 )b. ic. 1d. –i

____ 16. Evaluate the expression 40

2 i 35

2 i and write the result in the form a + bi.

a. 2 + 15ib. –15 – 2ic. 2 – 15id. 15 – 2i

____ 17. Find the polynomial P x( ) of degree 3 with integer coefficients, and zeros 3 and 2 i .

a. 3x 2 5x6

b. x 3 3x 2 4x3

c. x 3 5x 2 3x12

d. x 3 5x 2 6x

e. x 3 3x 2 4x12

____ 18. Find the x- and y-intercepts of the rational function r ( x ) =x 18x + 6

.

a. x-intercept (–18, 0), y-intercept (0, –2)b. x-intercept (–3, 0), y-intercept (0, 18)c. x-intercept (18, 0), y-intercept (0, –3)d. x-intercept (18, 0), y-intercept (0, –5)e. x-intercept (–1, 0), y-intercept (0, 18)

____ 19. Find the vertical asymptote of the rational function r ( x ) =x2 + 1x 9

.

a. x = 9b. x = 1c. x = 18d. x = 1e. x = 9

Name: ________________________ ID: A

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____ 20. Find the intercepts and asymptotes of the rational function r ( x ) =9x + 1084x + 12

.

a. x-int. y-int. horiz. asymptote vert. asymptote(–12, 0) (0, 9) y 4 x = –2.25

b. x-int. y-int. horiz. asymptote vert. asymptote(–12, 0) (0, 9) y 9 x = –9

c. x-int. y-int. horiz. asymptote vert. asymptote(0, –12) (9, 0) y 2.25 x = 3

d. x-int. y-int. horiz. asymptote vert. asymptote(–12, 0) (0, 9) y 2.25 x = 3

e. x-int. y-int. horiz. asymptote vert. asymptote(–12, 0) (0, 9) y 3 x = –2.25

____ 21. Find the y-intercept and asymptotes of the rational function r ( x ) =75

( x 5) 2 .

a. y-intercept horizontal asymptote vertical asymptote (0, 3) y = 5 x = 0

b. y-intercept horizontal asymptote vertical asymptote (0, 5) y = 0 x = 3

c. y-intercept horizontal asymptote vertical asymptote (0, 3) y = 0 x = 5

d. y-intercept horizontal asymptote vertical asymptote (0, 3) y = 1 x = 5

e. y-intercept horizontal asymptote vertical asymptote (0, 5) y = 1 x = 3

____ 22. Find the slant asymptote of the function y =x2

x 1.

a. y = x + 1b. y = x 3c. y = x + 5d. y = x + 4e. y = x 2

Name: ________________________ ID: A

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____ 23. Find the exponential function f(x) a x whose graph is given.

a. f(x) 4x

b. f(x) 4x 4

c. f(x) 4x

d. f(x) x4

e. f(x) 4x

____ 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. $1629b. $2653c. $1000d. $2753e. $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. $627b. $623c. $875d. $1605e. $1594

Name: ________________________ ID: A

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____ 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

Name: ________________________ ID: A

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a.

Domain: (4,). Range: (,).Asymptote: y 3.

d.

Domain: (3,). Range: (4,).Asymptote: y 3.

b.

Domain: (3,). Range: (,).Asymptote: x 3.

e.

Domain: (,). Range: (4,).Asymptote: y 4.

Name: ________________________ ID: A

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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) 12e0.011t

where m(t) is measured in kilograms. How much of the mass remains after 25 days?a. 9.02 kgb. 15.80 kgc. 12.76 kgd. 9.22 kge. 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 21.7e0.385t

where t is measured in years. What size does the population approach as time goes on?a. 2500b. 7500c. 5000d. 500e. 100

Name: ________________________ ID: A

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____ 29. Express the equation in exponential form.

log416 2

a. 24 16b. 42 16c. none of these

d. 216 4e. 162 4

____ 30. Express the equation ln (x + 1) = 4 in exponential form.a. none of these

b. x e1 4c. x e4 1d. x e1 4e. x e4 1

____ 31. Express the equation in logarithmic form.

10 3 = 1,000

a. log 3 10 = 1,000b. log 3 1,000 = 10c. log 10 1,000 = 3d. log 1,000 10 = 3e. none of these

____ 32. Express the equation in logarithmic form.

ex 2 0.2a. x = 0.2 – ln 2b. x = 0.2 + ln 2c. x = 2 + ln 0.2d. x = –2 + ln 0.2e. none of these

____ 33. Evaluate the expression.

e ln 5

a. 5eb. 5c. none of thesed. ln5e. e5

Name: ________________________ ID: A

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____ 34. Evaluate the expression.

10log

a. b. logc. none of these

d. 10

e. 1

____ 35. Use the definition of the logarithmic function to find x.

logx 81 4

a. x = 81b. none of thesec. x = 5d. x = 4e. x = 3

____ 36. Find the function of the form y loga x whose graph is given.

a. y log8(x)

b. none of thesec. y log2(x)

d. y log3(x)

e. y log5(x)

Name: ________________________ ID: A

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____ 37. Use the graph of y = log 3 x below to help you identify the graph of y = 3 x .

a. d.

b. e. none of these

c.

Name: ________________________ ID: A

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____ 38. Find the domain of the function.

f(x) x 5 log3(11 x)

a. [–11, –5)b. none of thesec. [5, 11]d. [–5, 11]e. [5, 11)

____ 39. Find the domain of the function.

f(x) log7 x x10ÊËÁÁÁ

ˆ¯˜̃̃

a. (–1, 1)b. none of thesec. (1,)d. (0,)e. (0, 1)

____ 40. Evaluate the expression.

log 3 189 – log 3 7a. 21b. ln 189c. 7d. 3e. 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.

log7 x2 58

a. 8log7 x2 5ÊËÁÁÁ

ˆ¯˜̃̃

b. log7

x2 58

c.18

2log7s log75ÊËÁÁ

ˆ¯̃̃

d. log7 x2 5ÊËÁÁÁ

ˆ¯˜̃̃

e.18

log7 x2 5ÊËÁÁÁ

ˆ¯˜̃̃

Name: ________________________ ID: A

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____ 42. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.

ln 3r8s9

a.89

ln3 89

lnr 89

lns

b.19

ln3 89

lnr 19

lns

c. ln3 89

lnr 19

lns

d.19

ln3 19

lnr 19

lns

e. ln3 89

lnr lns

____ 43. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.

loga 6

b 3 c

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

a. 6loga 3logb 12

logc

b. 6loga 3logb 12

logc

c. 6loga 3logb logc2

d. 6loga logb 12

logc

e. log(6a) 3logb 12

logc

Name: ________________________ ID: A

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____ 44. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.

ln xyz

9

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃̃

a. lnx 19

lny 19

lnz

b. lnx 19

lny 19

lnz

c. lnx 19

lny 19

lnz

d.19

(lnx lny lnz)

e. lnx 19

lny 19

lnz

____ 45. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.

log x y z666

a.1

216logx 1

36logy 1

6logz

b.16

logx logy logzÊËÁÁ

ˆ¯̃̃

c.16

logx 136

logy 1216

logz

d.1

216logx logy logzÊ

ËÁÁˆ¯̃̃

e.1

216logx 1

36logy 1

6logz

____ 46. Rewrite the expression below as a single logarithm.

log14 12

log3 log2

a. log17

3

b. ln3 7

c. log13

7

d. log21

e. log7 3

Name: ________________________ ID: A

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____ 47. Find the solution of the exponential equation.

e42 x = 14

a. x = 1.8105b. x = 0.6805c. x = –0.0963d. x = 2.9391e. x = 2.7183

____ 48. Find the solution of the exponential equation, correct to four decimal places.

12x 5x 4

a. x = 7.3547b. none of thesec. x = 1.544d. x = 7.3535e. x = 0.544

____ 49. Find the solution of the exponential equation, correct to four decimal places.

1.008085x 8a. x = 0.625b. x = 1.6c. x = –1.6032d. x = –51.679e. x = 51.679

____ 50. Solve the equation.

e2x 5ex 4 0a. x = 4, x = 1b. x = 1.3863, x = 0c. x = 1.6094d. x = 0.7213, x = 0e. x = –4, x = 1

____ 51. Solve the equation.

e 2 x 8e x + 7 = 0

a. x = 2.0794b. x = –7, x = 1c. x = 0.5139, x = 0d. x = 7, x = 1e. x = 1.9459, x = 0

Name: ________________________ ID: A

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____ 52. Solve the logarithmic equation for x.

log 3 (4 – x) = 7a. x = –2183b. x = 2191c. x = 2187d. x = –2191e. x = –2187

____ 53. Solve the logarithmic equation for x.

log 2 2 + log 2 x = log 2 3 + log 2 (x – 5)a. x = 12b. x = 30c. x = 15d. x = 17e. x = 3.9

____ 54. For what value of x is the following true?

log (x + 9) = log x + log 9a. x = –7.875b. x = 0.051c. x = 4.5d. x = 1.125e. x = 10

____ 55. Solve the inequality.

log (x – 2) + log (9 – x) < 1a. 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.

x2ex 16ex 0

a. x (4, 0)b. x (0, 4)c. x (4, 4)d. x (4, 16)e. x (16, 16)

Name: ________________________ ID: A

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____ 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 yearsb. none of thesec. 12 yearsd. 50 yearse. 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 theseb. 14.65 yearsc. 0.09 yeard. 9.24 yearse. 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) 13e0.089t

where m( t ) is measured in grams. After how many days is there only 10 g remaining?a. 2 daysb. 6 daysc. 3 daysd. 5 dayse. 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 yrb. 54.14 yrc. 41.23 yrd. 108.28 yre. 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 sb. 44.94 sc. 1.54 sd. 129.66 se. 89.87 s

Name: ________________________ ID: A

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____ 63. An unknown substance has a hydrogen ion concentration of

HÈ

ÎÍÍÍÍÍ

˘

˚˙̇̇˙̇ 6.1 103M .

Find the pH.a. pH = 2.2b. pH = 12.9c. pH = 3.0d. pH = 5.1e. none of these

ID: A

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Test 2 Review (Math1650, $3.3-3.7 & Chap.4)Answer Section

MULTIPLE CHOICE

1. ANS: A2. ANS: C3. ANS: C4. ANS: D5. ANS: C6. ANS: B7. ANS: C8. ANS: C9. ANS: E

10. ANS: E11. ANS: B12. ANS: A13. ANS: C14. ANS: B15. ANS: C16. ANS: C17. ANS: E18. ANS: C19. ANS: E20. ANS: D21. ANS: B22. ANS: A23. ANS: C24. ANS: B25. ANS: B26. ANS: E27. ANS: E28. ANS: A29. ANS: B30. ANS: E31. ANS: C32. ANS: D33. ANS: B34. ANS: A35. ANS: E36. ANS: C37. ANS: C38. ANS: E39. ANS: D40. ANS: D

ID: A

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41. ANS: E42. ANS: B43. ANS: A44. ANS: A45. ANS: C46. ANS: E47. ANS: B48. ANS: D49. ANS: E50. ANS: B51. ANS: E52. ANS: A53. ANS: C54. ANS: D55. ANS: E56. ANS: C57. ANS: C58. ANS: D59. ANS: C60. ANS: A61. ANS: D62. ANS: D63. ANS: A