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Math 117 Practice Problems for Test III
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1. Factor fully each of the following:
(a) x3 + x2 – 7x + 2 given that x = 2 is a root
(b) x4 + 6x3 – 68x2 – 150x – 77 given that x = -1 is a root of multiplicity 2
(c) x3 – 8 given that x = 2 is a root
(d) x5 + 1 given that x = -1 is a root
2. Solve for x. Give an exact answer (without using calculator). Show your
work!
(a) 83x = 1024
(b)
(1/3)x = 243x+1
(c)
56x-3 = 125
3. Find an equation of a polynomial that has zeroes at x = 1, 4, 5 and has yintercept of 11.
4. Write (1/9)5 2434 33 as a power of 3.
5. (a) Without actually dividing, explain why x + 3 is a factor of
x3 + 6x2 + 11x + 6
(b)
Explain why x – 1 is a factor of 32x74 – 33x33 + 1
6. Multiply the following two polynomials:
p(x) = x4 + 3x3 – 4x2 + x +1
q(x) = x6 – x5 + x3 + 3x2 +1
7. Consider the polynomial
y = f(x) = –x2(x – 2)4(x – 3)5(x – 5)(x2 + x + 1)
(a) The domain of f is:
(b) The zeroes of the polynomial are:
(c) What happens to y as x → ∞ ?
(d) What happens to y as x → -∞ ?
8.
Consider y = f(x), a rational function that has the graph below:
(a) List the zeroes.
(b) List the singularities.
(c) List the horizontal asymptotes.
(d) List the vertical asymptotes.
9. Simplify each of the following and express with positive exponents:
( A)
.
3

ab1c 2 (a 1b 2c 4 ) 1/ 6
3
 a 2b 
 3 4 
ab 
( B) 
5
 ab 1 
 3 2 
a b 
10. Perform polynomial division. What is the quotient? What is the remainder?
x 3  3x 2  x  4
(a)
x 1
x3  x 2  x  4
(b)
x2  x  1
11. Find the quotient when x3 – 4x2 + 5x + 6 is divided by x – 2.
12. What are the roots of the polynomial p(x) = x3 – 2x2 – 23x + 24 ?
Hint: Since p(1) = 0, x = 1 is a root of this polynomial.
13. Express 0.0001024 in the form 2a 5b .
14. For which value(s) of c will the following polynomial be divisible by x – 3?
P(x) = 3x4 – cx2 + 3x – 72
15. Solve for x: (a) 43x+1 = 162-5x
(b)
125x-4 = (1/25)3x-1
(c)
3x 92x+5 = (1/81)3-9x
16. Consider the following two rational functions:
r ( x) 
x
2
and s( x) 
2x  3
x 4
2
Express each of the following as a rational function. Simplify.
(a) r(x) + 3 s(x)
(b) (s(x))2
(c) (rs)(x)
(d) (r  s )( x)
(e) ( s  r )( x)
17. If the population of owls in Hogwarts is currently 79 and tripling every 8
months, and the population of mice is currently 15,023 and falling 20% every 7
months, when will the number of owls equal the number of mice?
18. Suppose that Harry’s niece, Albertine, is born today. Harry would like to give
Albertine a check for $50,000 when she turns 18 years old. How much should
Harry deposit today at the LUC bank given that the bank pays annual interest of
3.78 % compounded weekly?
19. If log2 3 = a and log2 5 = b, express log2 25/162 in terms of a and b.
20. Solve for x: log81(log3 x) = -1/2.
21. Solve for t: log13 t + log13 (12t – 23) = log13(24).
22. Solve for y: (log10 y)2 – 14 log10 y = -45.
23. Radioactive isotope XZX decays from 300 mg to 85 mg in 5 years. Find its
half-life.
24. True or False: (Assume that a and b are positive real numbers.)
(a)
log8(ab) = log8 a + log8 b
(b) log7 (a+b) = log7 a + log7 b
(c) log10 (0.0001) + log4 256 = 0
(d) log130 (a/b) = (log130 a)/ (log130 b)
(e) log10 (1451) = (log10 14)51
(f) log3 10 = 0
(g) 5x > 4.9x for all x > 0
(h) log10 (1 / 3.19) = - log10 3.19
(i) log4 7 = (log2 7)/2
25. Suppose that a radioactive isotope loses 1/5 of its atoms in a sample decay after
14 years. Find the half-life of the isotope.
26. Solve for x:
27.
9 – 2(log2 x – 1) = 5(2 – log2 x) + 1.
100a 2 (bc)13
Express the logarithm of
in terms of log10 a, log10 b and log10 c.
c5 b
28. Solve for x in terms of a, b, c:
a
b
a b


x a x b x c
29.
(a) If log5 2 = a, log5 3 = b, and log5 7 = c, express log5 (2000/1323) in
terms of a, b and c.
(b)
Solve for x:
3 log 10 x  log 10 (6 x 3  1)  1
30. Solve the inequality log3( 2x – 5) ≤ 3
31. Suppose that a radioactive isotope loses 17% of its initial amount in 41 years.
Find the half-life of the isotope.
32. How many digits are in the number 52016 ?
33. True or False:
(a)
If A and B are positive real numbers, then (logB A)(logA B) = 1
(b) 2log2 (5 / 7 )  5 / 7
____________
(c) The domain of log10 (7 – 2x) is (-∞, 7/2) _________
(d)
log13 15 = log15 13
____________
(e)
(ln 7)3 = 3 ln 7
(f)
log1/2 71 < log1/2 13
_________
______________
(g) The inverse of f(x) = e3x – 7 is given by g(x) = (7 + ln x)/3.
34. Sketch the curves y = 2x, y = 3x , y = ex and y = (1/2)x on the same pair of
axes. Indicate all points of intersection.
35.
Find the domain of the function G(x) = ln(x – 4) + ln(7 – x) + ex
36. In the year 2013, Alphaville had 4000 inhabitants and Betaville had only
2000. The population of Alphaville is declining with a half-life of 15
years. The population of Betaville is growing at a rate of 33% every 4
years. When will the population sizes of Alphaville and Betaville coincide?
(Give the answer to the nearest year.)
37. Suppose that 2 mg of a drug is injected into a patient’s bloodstream. As the
drug is metabolized, the quantity diminishes at the rate of 4% per hour.
(a) Find a formula for Q(t), the quantity of the drug remaining in the body
after t hours.
(b) By what percent does the drug level decrease during any given hour?
(c) The patient must receive an additional 2 mg of the drug whenever its level
has diminished to 0.25 mg. When must the patient receive the second
injection?
(d) When must the patient receive the third injection?
38. (a) Write the following expression as a sum and/or difference of
logarithms. Express powers as factors.
ln
x x2  5
( x  3)11
(b) Write the following expression as a single logarithm.
log 3 (5x  11)  2 log 3 (2 x  5)  log 3 x  5 log 3 9
39.
(a) Solve the following equation for x. Express your answer to the nearest
hundredth.
19(0.81) 3 x1  2 x1 (1.31) 2 x
(b) Solve for t:
log10(509 t – 11) – log10 (t – 3) = 3
40. Traces of burned wood along with ancient stone tools in an archaeological dig
in Chile were found to contain approximately 1.67% of the original amount
of carbon 14. If the half-life of carbon 14 is 5600 years, approximately
when was the tree cut and burned?
41. How many digits are there in the number 4599777888 ? Express the number in
scientific notation.
42. A website received 1900 hits on January 1, 2015. The number of hits per day
is growing exponentially; it increases by 7.4% per day. Find a formula for
the number of hits per day for this website and use your formula to predict
the number of hits on July 4, 2015.
43. Find the inverse of the function y = ln(3 – 7x) + 1.
44. Express the number 70709991 in scientific notation.
45. Estimate the value of:
10100
4 

1  100 
 10 
46. Find a polynomial p(x) that has roots 1, 2, -3, -4 and satisfies the property that
p(-1) = 5.
47. Technetium-99m is a radioactive substance used to diagnose brain diseases.
Its half-life is approximately 6 hours. Initially a patient is given 200 mg of
technetium-99m.
(a)
Write an equation that gives the amount, A(t), of technetium-99m
remaining after t hours.
(b)
Determine the number of hours needed for the initial dose to decay to 35
mg.
48. How many digits are there in the number 99999999 ?
49. Solve for x:
2(log3 x)2 + 5(log3 x) – 12 = 0
Hint: First let y = log3 x and solve for y. Then find x.
50. Let f(x) = x/(x + 2), and let g(x) = 1 + 4x2.
(a) Calculate
f ( x  h)  f ( x )
. Simplify your answer as much as possible
h
(b) Calculate
g ( z  k )  g ( z)
. Simplify your answer as much as possible.
k
51. Find the vertex of the given parabola:
(a) y = x2 – 14
(b) y = 5x2 + 15x + 13
(c) y = -9x2 + 3x – 44
(d) y = (x – 7)2 + (x – 4)2
(e) y = 3(x + 8)2 + 5x
(f) y = x(x – 2) + x – 1
52. Find the vertex of each of the following parabolas. Also, sketch each curve
and locate the y-intercept, any x-intercepts, and label the vertex. In addition,
determine the range of the function.
(a) y = x2 – 8x + 1
(b) y = -x2 + x + 8
(c) y = 4x2 – 12x – 13
53. Solve each of the following equations by completing the square or explain
why no solution exists:
(a) x2 – 6 x + 2 = 0
(b) 4x2 + 4x + 1 = 0
(c) 5x = x2 + 9
54. Explain the significance of the discriminant of a quadratic expression
Ax2 + Bx + C. Give examples of each of the three types of discriminants
and their relationship to the corresponding graph of the parabola.
55.
56.
Find two numbers whose sum equals 10 and whose product equals 7.
How many real roots does each of the following equations possess? (Hint:
These questions require very little calculation.)
(a)
(x – 1) (x + 5) (x2 + 13)4 = 0
(b)
5x2 – 4x + 1 = 0
(c)
x2 – 4x – 1 = 0
(d)
3x2 – 4x + 8 = 0
(e)
(x4 + 2)(x + 1)(x2 – 9) = 0
(f) Show that there do not exist two real numbers whose sum is 7 and
whose product is 13.
(g) Find a constant c such that the graph of the parabola y = x2 + 5x + c
has its vertex on the line y = x.
57. Consider the circle (x + 11)2 + (y + 999)2 = 4. If the radius is increased by 5
cm, how much does the area change? Give an exact answer, using appropriate
units.
. Simplify and express with positive exponents:
 ab c  (a b
3
1 2
1 2 4 1/ 6
c )
58. Simplify fully and express with positive exponents:
3
 a 1bc 
 3 4 
a b 
2
 ab 1c 
 3 3 
a b 
59. Suppose that the graph of f is a parabola with vertex at (7, 11).
60. Let g(x) = 1 – 3x. Find the vertex of the parabola f  g .
61. Suppose that the graph of f is a parabola with vertex at (1, 3).
62. Let g(x) = 4x + 5. Find the vertex of the parabola y  g  f .
63. Simplify fully (and express your answer using only positive exponents):
a bc
2
ab 2 c 9
3
abc 4
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- Ralph Waldo Emerson