2nd Semester Alg 2
Name
1 - Quadratic Inequalities, Complex Numbers, 6.1, Binomial Expansion
Graph each inequality.
1. y  x 2  4 x  3
2.
y  2 x 2  x  6
Solve the quadratic inequalities algebraically. Answers in interval notation.
3. x 2  10 x  17  4
5. x 2  16  9  10 x
4.
x 2  6 x  36  36
6.
x3  2 x 2  x  2  0
Simplify the following.
7.
 2  i  2  i 
10.
4i
2i
8.
5  3i  4  2i 
11.
2  2i
5  2i
9.
1  4i 5  7i 
12.
7i
9i
Find each product.



13.  x  3 x 2  2 x  1
Expand the following binomials.
15.  c  3d 
5
16.
Find the indicated term in each binomial expansion below.

3
2
17. the 4th term of d  3e

14.  4 x  5  2 x5  x 3  1

5
 3x  2 y 
3 4

6
18. the 7th term of 3x  2 y

6
Use long division to find the quotient.



19. x 2  3 x  9   x  5 

20. 8 x 4  4 x 2  x  4   2 x  1
2 - Polynomials and Synthetic Division
Use the sum and product to find the polynomial equation with the given solutions.
1.
2,
2
3
2. 3  2i
Divide by long division.
4. 6a 3  11a 2  4a  9   3a  2 

1  3i
2
3.

 27 x
5.
3
 8   3x  2 
4
 5 x 2  10 x  12    x  2 
Divide by synthetic division.
6.  x 4  3x 2  6    x  2 
7.
x
Is the first polynomial a factor of the second?
8. x  1; x3  5x 2  6 x  2
9.
x  3; x3  7 x  6
Given one factor, factor completely.
10. x  3; x3  3x 2  x  3
11. x  4; 2 x3  42 x  40  0
Solve each of the following with a calculator.
(Hint: you will have to use the quadratic formula on some of them!)
12. 4 x 4  13x3  15 x 2  7 x  1  0
14. x4  4x3  3x2 14x  8  0
13. x 4  10 x3  35 x 2  50 x  24  0
15. x3  7 x2  14x  6  0
Evaluate P  x  for the given value.
16. P  x   2x4  26x2  3x  6 find P  4
17. P  x   x5  x4  x3  3x2  1 find P  2
Write a polynomial with the given factors.
18. x  4, x  4, 4 x  1
19. x  1, x  2, x 
Name the possible solutions. Do Not solve
21. 2x4  4x3  3x2 14x  8  0
23. Find k so that x – 2 is a factor of x3  4x2  kx  12
3
2
20. x  5, x  3i, x  3i
22. 3x3  4x2  4x 12  0
3 - Logs 1
For each of the following, graph the function and its inverse on the graph provided. Name the equation of the
inverse.
2. Graph the inverse of the equation pictured
1. g ( x)  2 x
below.
g ( x)1 =_____________



























Rewrite each equation in logarithmic form. (No calculator).
3. b x  k
2
4. 9 3  27
Rewrite each equation in exponential form. (No calculator).
5. log8 x  2
6. logb t  p
Evaluate each logarithm. (No calculator).
7. log10000
12. log b b d
8.
log 1
7
1
49
log 4
1
16
10. log 64
1
2
9.
11. log8 4 log 2 32
(Expand) Write each expression in terms of
17. log 4 c2b3


13. log 5
1
125
14. log 1 8
2
15. log 3
2
27
8
16. log 8 1
log 4 a, log 4 b, and log 4 c .
3
18. log a b
4
2
c
(No calculator)



19. Write 2log b m  4log b n  3log b
p as a logarithm of a single number or expression. (No calculator)
Solve each by making the bases alike.
20. 43 2 x  32 x
1
21.  
2
Solve the following equations. (No Calculator).
23. log 5 (1  3 x)  2
4 – Logs 2
8
2 x 3
22.  32 
3x
 1
 
 16 
2 x 1
24. log 2  x  2   log 2  3x  1  4
25. log  x  1  log x  log12
A  Pe rt
3 x 1
26. 3log x  log x  log9
 r
A  P 1  
 n
nt
Write an equation and solve.4
1. You invest $3000 at 8% compounded monthly. How long will it take you to earn $20,000 interest? (Round to the
nearest year).
2. Bob invests $5000 with dreams of becoming a millionaire. If his investment earns 9% compounded daily, how
long will it be until Bob’s dreams come true? (Round to the nearest year).
3. If you invest $4500 dollars for 35 years at 8% compounded monthly, how much money will you have?
Solve for x. Round to 4 decimal places as necessary.
4. 344  242 x1
5.
242 x3  332
6.
log2 x  log2  x  6  4
7.
log4  x  2  log4  x  2  1
8.
ln 3.2981  x
9.
log  2 x  4   1
10. x  log3 324
11. log4  3x  2   2
12. e2 x  2  3
Simplify without a calculator.
13. 2 log 2 2  3log 3
1
9
14. 5log100  3log 2 4
15. y varies inversely as x: y  9 when x  15 .
Find y when x  36.
Equation_______________ y = _________
16. y varies directly as x: y  24 when x  6 .
Find y when x  4.
Equation_______________ y = _________
5-Rational Expressions
Simplify each
1.
9 x 2 yz
24 xyz 2
2.
x 2  2 x  15
x 2  x  12
3.
25  x 2
x 2  x  30
5.
6r  3 r 2  9r  18
r 6
2r  1
6.
x
x2
 2
x  2 x  5x  6
Multiply or divide each and simplify.
4.
m2  8m  15 7a  14b
a  2b
m3
Add or subtract, and simplify if necessary.
7.
2 1 3
 
x 4 2x
8.
4x 1
3

2
x 4 x2
9.
3  4x
2

x  3 x  10 x  5
10.
3 x
4

2
x  9 2x  6
11.
5x
4
 2
x  x  6 x  4x  4
12.
5x
4
9


x  7 x 2 x  14 x
14.
x 3
4

1
x  5 10  2 x
15.
x 1
x

x  2 x 1
Solve for x and name any restrictions on x.
13.
16  x
2
5 x
16.
x
3
 4
x 3
x 3
2
17.
2
2
1
8

 3
x  2 x 1
Simplify
3 2

x y
19.
4 5

3x y
2
3

18. x  9 4 x  12
1
5

4 2x  6
2
6 - CIRCLES, ELLIPSES
For each circle, identify the center and the radius.
1.
 x  3 2   y  5 
2
 16
2.
x 2  2 x  1  y 2  12 y  36  144
3. Write the equation of the circle with cente (3,1) and radius 4.
4. Write the equation of the circle with diameter
endpoints (-2,-3) and (4,1).
5. Rewrite the equation for the circle below in standard form and identify its center and radius.
4x2  4 y2  32 x 16 y  71  0
Write the equation for the circles graphed below.
6.
7.
































8. Find the value of k so that the given points are n units apart given (2, -2) and (-3, k); n = 13
9-11 Use the given information to write an equation of an ellipse in standard form.
9. endpoints of minor axis (2,0) (2,0), major axis 10
units long
12. Equation: ________________
10. foci (2,8) and (2,-2, minor
axis length 18
11. center (3,1), horizontal,
major axis length 10;
minor axis length 6
13. Equation:________________
Center ____________
Center ___________
Vertices: ______ ______
Vertices: ______ ______
Foci: ______ _______
Foci: ______ ______
Major axis: ________
Major axis: ________
Minor axis:________
Minor axis:_______
Graph each. Also give the foci, vertices, and lengths of the major and minor axes.
14. .  x  1   y  2   1
12
16
2
2
15. 4 x 2  16 x  25 y 2  50 y  59  0
Center ____________
Center ____________
Vertices: ______ ______
Vertices: ______ ______
Foci: ______ _______
Foci: ______ _______
Major axis: ________
Major axis: ________
Minor axis:________
Minor axis:________
16. 2 x 2  y 2  8x  2 y  5  0
Center ____________
Vertices: ______ ______
Foci: ______ _______
Major axis: ________
7 - Parabola/Hyperbolas
Graph each parabola. Fill in each blank.
1.
V
y 1 
1
2
 x  2
4
F
2. x  2  
C=_______
V
1
2
 y  1
12
F
C=_______
Graph and find the equation of the parabola if
3. the vertex is (3,0) and the directrix is y = -1.
F(
)
C=_______
4. the vertex is (2,-7) and focus is (
dir:
Equation:
7
,-7).
4
C=_______
Equation:
Find the equation of each parabola.
5.
6.
Equation:______________________
Equation:______________________
V:________ F:________
V:_________
dir:________
F:__________ dir:__________
Graph each. Also give the foci, vertices, and equations of the transverse and conjugate axes.
7.
y 2 x2

1
4 25
2
2
8. x  y  1
49 36
center:___________
center:___________
Vertices: ______________
Vertices: ______________
Foci: _____________
Foci: _____________
trans axis: _____
trans axis: _____
conj axis: _______
conj axis: _______
Asymptotes:
Asymptotes:
 x  4
9.
36
2
 y  1

2
81
1
10. y 2  16 x 2  8 y  32 x  64
center:___________
center:___________
Vertices: ______________
Vertices: ______________
Foci: _____________
Foci: _____________
trans axis: _____
trans axis: _____
conj axis: _______
conj axis: _______
Asymptotes:
Asymptotes:
Write an equation for the hyperbola with the given characteristics.
11. center (1,3); vertex (7,3); focus (10, 3)
12. asymptotes: y   3 x ; foci (0,10) and (0,-10)
4
Identify the conic.
13. 4 x 2  y 2  8x  6 y  9  0
14. x 2  8x  7  9 y 2  0
15. x 2  4 y  6 x  17  0
16. 4 x 2  4 y 2  24 x  16 y  9  0
8 - Series and Sequences
Find the indicated term.
1. 2, 6,18, 54,...a13
2. 12, 1, 14,...a100
1
4
Find the sum.
5. 100  79  58  37  ...S150
64 64

 ...S7
3
9
7. 100  96  92  88  ...S200
8. 5  10  20  40  ...S8
6.
192  64 
3. 16, 4,1,  ,...a12
4.
4,15,26,37,...a 250
Find the infinite sum, if it exists.
9. 5  9  13  17  ...S

n 1
3
10.  8   
5
n 1
11. 120  60  30  15  ...S
Write each in sigma notation.
12. 8  29  50  71  ...S150
1
100  

2
n 1
13
15.
13. 400  200  100  50  ...S10
n 1
16. Find 3 geometric means between 400 and
25/16.
Find the sum from its sigma notation.
200
14.
 3n  2
17. Find 4 arithmetic means between 6 and 58.
n 1
9 - Graphing Polynomial and Rational Functions
Graph each.
1.
f  x   x  x  2   x  6   2 x  5
3
2
5.
2.
f  x     x  2  x  4   x  5 
2
2
6.
3.
4.
f  x 
f  x 
2
x2
7.
3
x  x  3
2
8.
 x  2  x  2 
2
 x  1 x  3
 x  1 x  1
f  x 
 x  4  x  2 
2
x  3  x  2 

f  x 
 x  2  x  1 x  5
f  x 
f  x 
x2  5x  6
 x  1