Download College Algebra Review Part A

College Algebra Review Part A
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Rewrite the expression as a sum or difference or multiple of logarithms. Expand.
x5 y7
1) log3
8
Answer: 5 log3 x + 7 log3 y - log3 8
2) log 25x
Answer: 2log 5 + log x
3) ln x4y4
Answer: 4ln x + 4ln y
4) log
4 x
y
Answer: 1 log x - 1 log y
4
4
Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm.
Assume that all variables represent positive real numbers. Condense.
5) log5 6 - log5 m
Answer: log5 6/m
6) logb x + logb y
Answer: logb xy
7) 4 log5 (2x - 1) + 5 log5 (6x + 1)
Answer: log5 (2x - 1)4 (6x + 1)5
8)
2
1
logax + loga y
5
9
Answer: loga (x2/5 y1/9)
9)
1
logax + 3 loga y - 4 loga x
2
3
Answer: loga y
x7/2
1
Solve the exponential equation. Express the solution set in terms of natural logarithms.
3x
= 2.2
10) 8
Answer: ln 2.2
3 ln 8
11) 3
x+7
=4
Answer: ln 4 - 7
ln 3
12) e2x = 5
Answer: ln 5
2
13) e
x+6
=3
Answer: {ln 3 - 6}
Solve the logarithmic equation. Be sure to reject any value of x that produces the logarithm of a negative
number or the logarithm of 0.
14) log 3 x = 4
Answer: {81}
15) log 2 (x - 3) = 1
Answer: {5}
16) log 3 (x - 4) = -2
Answer: { 37 }
9
17) log 4 (x + 3) + log 4 (x - 3) = 2
Answer: {5}
18) log 2 (x + 2) = 1 + log 2 (x - 5)
Answer: {12}
19) log 10 (x - 30) = 3 - log 10 x
Answer: {50}
2
Solve the equation by isolating the natural logarithm and exponentiating both sides. Express the answer
in terms of e.
20) ln x = 3
Answer: {e3 }
21) 6 ln 2x = 18
3
Answer: e
2
22) 6 + 6 ln x = 11
Answer: {e 5/6 }
23) ln x + 9 = 9
Answer: {e18 - 9}
The graph of a quadratic function is given. Determine the function's equation.
24)
Answer: f(x) = (x + 2)2 + 2
Find the coordinates of the vertex for the parabola defined by the given quadratic function. Find axis of
symmetry.
25) f(x) = (x - 5)2 - 5
Answer: (5, -5)
26) f(x) = (x + 8)2 - 4
Answer: (-8, -4)
27) f(x) = x2 + 14x + 3
Answer: (-7, -46)
3
28) f(x) = -x2 + 12x - 7
Answer: (6, 29)
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
29) f(x) = -6x2 + 12x + 5
Answer: (1, 11)
Use the vertex and intercepts to sketch the graph of the quadratic function.
30) y - 1 = (x + 2)2
Answer:
4
31) f(x) = 3(x - 2)2 - 3
Answer:
5
32) f(x) = x2 - 4x + 3
Answer:
Divide using long division.
33) (x2 + 15x + 56) ÷ (x + 7)
Answer: x + 8
34) (5x2 + 17x - 40) ÷ (x + 5)
Answer: 5x - 8
35)
-6x3 + 25x2 - 35x + 25
-2x + 5
Answer: 3x2 - 5x + 5
Solve the problem.
36) A rectangle with width 2x + 3 inches has an area of 2x4 + 7x3 - 16x2 - 57x - 36 square inches.
Write a polynomial that represents its length.
Answer: x3 + 2x2 - 11x - 12
6
Divide using synthetic division.
3x2 + 29x + 56
37)
x+7
Answer: 3x + 8
38)
5x3 - 34x2 + 29x - 30
x-6
Answer: 5x2 - 4x + 5
39)
x5 + x3 + 5
x-2
Answer: x4 + 2x3 + 5x2 + 10x + 20 + 45
x + -2
Solve the problem.
40) Given f(x) = 4x3- 6x2 - 5x + 6, use the Remainder Theorem to find f(-3).
Answer: f(-3) = -141
41) Given f(x) = 7x4 + 9x3 + 4x2 - 4x + 16, use the Remainder Theorem to find f(2).
Answer: f(2) = 208
Use the Rational Zero Theorem to list all possible rational zeros for the given function.
42) x5 - 3x2 + 6x + 14
Answer: ± 1, ± 7, ± 2, ± 14
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the
given function.
43) f(x) = -7x9 + x5 - x2 + 5
Answer: 3 or 1 positive zeros, 2 or 0 negative zeros
44) f(x) = 6x5 - 5x4 + x + 7
Answer: 2 or 0 positive zeros, 1 negative zero
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
45) f(x) = x3 - 6x2 + 9x - 4
Answer: {1, 4}
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Answer Key
Testname: UNTITLED1
1) 5 log3 x + 7 log3 y - log3 8
2) 2log 5 + log x
3) 4ln x + 4ln y
1
1
4) log x - log y
4
4
5) log5 6/m
6) logb xy
7) log5 (2x - 1)4 (6x + 1)5
8) loga (x2/5 y1/9)
y3
9) loga
x7/2
10)
ln 2.2
3 ln 8
11)
ln 4
-7
ln 3
12)
ln 5
2
13) {ln 3 - 6}
14) {81}
15) {5}
37
16) { }
9
17) {5}
18) {12}
19) {50}
20) {e3 }
e3
21)
2
22) {e 5/6 }
23) {e18 - 9}
24)
25)
26)
27)
28)
29)
f(x) = (x + 2)2 + 2
(5, -5)
(-8, -4)
(-7, -46)
(6, 29)
(1, 11)
8
Answer Key
Testname: UNTITLED1
30)
31)
32)
33) x + 8
34) 5x - 8
35) 3x2 - 5x + 5
36) x3 + 2x 2 - 11x - 12
37) 3x + 8
38) 5x2 - 4x + 5
39) x4 + 2x 3 + 5x2 + 10x + 20 +
45
x + -2
40) f(-3) = -141
9
Answer Key
Testname: UNTITLED1
41)
42)
43)
44)
45)
f(2) = 208
± 1, ± 7, ± 2, ± 14
3 or 1 positive zeros, 2 or 0 negative zeros
2 or 0 positive zeros, 1 negative zero
{1, 4}
10