Download Test Review: Rational Functions and Complex Zeros

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Date: ___________
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Test Review: Rational Functions and Complex Zeros
1. Simplify and write the answer in standard form.
a. (3  4i )  (2  5i)
1  9i
b. (2  i)  (3  2i)
1  3i
c.
(6  i )(2  3i)
15  16i
d.
2  3i
4i
3 1
 i
4 2
2. Write the complex conjugate of the following:
a. 3  5i
3  5i
b. 3  4i
3  4i
c.
3  4i
3  4i
d. 6i
6i
3. What is the quadratic formula? (memorize it!)
x 
b  b 2  4ac
2a
4. Use the quadratic formula to solve:
a.
7 x 2  50
b. 2 x 2  6 x  4
x 
5 14
7
x  2,1
5. A polynomial of degree 4 has how many zeros?
6. The function f ( x)  7 x8  5x 6  3x 2  1 has how many zeros?
4
8
7. If a 5th degree polynomial has zeros at 3, 2+4i, and -3-5i, what are the remaining zeros?
2  4i ,
 3  5i
th
8. If a 4 degree polynomial has zeros at 2, 5i, and 1, what are the remaining zeros?
5i
9. The polynomial f ( x)  x3  5 x 2  16 x  80 has the zero 4i. Find the remaining zeros.
5, -4i
10. The polynomial f ( x)  x3  2 x 2  81x  162 has the zero -9i. Find the remaining zeros.
9i, 2
11. Use synthetic division to determine if x-4 is a factor of f ( x)  x3  2 x 2  23x  60 .
Yes, you should get a remainder of zero.
12. Simplify the following and note any domain restrictions:
2x2
4x
x4
b.
2
x  x  20
6
2

c.
x x4
1
6

d.
2 x4
( x  2)( x  3)2 ( x  4)
e.
( x  2)( x  3)( x  5)
a.
x
2
x 0
1
x  5, 4
x 5
4x  24
x  0, 4
x 2  4x
x  16
x  4
2x  8
(x  3)(x  4)
x 5
x  2, 3,5
13. Solve the following and list any extraneous solutions.
2 3 x

x
5
x  4 2x  8
 2
b.
x
x x
a.
Solutions: __2, -5_______ Extraneous Solutions: ___none___
Solutions: __-4, 3______ Extraneous Solutions: __0_______
14. Vertical Asymptotes are determined by look at: ___The denominator after you reduce._____
15. How are holes in the graph found? _You see what factors cancel out COMPLETELY from the
denominator and set those factors equal to zero. ________________
16. Find all asymptotes and discontinuities (holes) in the following rational functions.
a.
f ( x) 
( x  1)( x  2)
( x  2)( x 2  1)
Holes: _x = 2, -1___
b.
HA: _y =0__________
x2  6 x  8
f ( x)  2
( x  4)( x  6)
Holes: _x= -2_______
c.
VA: _x =1___________
f ( x) 
VA: _x=2, x=6___
HA: _y = 0_______
2x  6
x  x  12
2
Holes: _x = -3______
VA: _x = 4 _____
HA: _y = 0_______
17. Determine the Horizontal asymptote of each function:
a.
b.
c.
2 x3  x 2  8
f ( x) 
x4
( x  5)( x  6)
f ( x)  2
x  3x  1
( x  4)( x  6)
f ( x) 
( x  2)( x  1)( x  3)
HA: _none________
HA: __y = 1______
HA: _y = 0_______
18. Write a possible equation, of least degree, of a rational function with a hole at x = 5, and vertical
asymptotes at x = 4 and x = -2.
f (x ) 
(x  5)
(x  5)(x  4)(x  2)
19. Write a possible equation, of least degree, of a rational function with a holes at x = 3 and x = 2,
and vertical asymptotes at x = 5 and x = -2
f (x ) 
(x  3)(x  2)
(x  3)(x  2)(x  5)(x  2)
20. Identify the right-hand and left-hand behavior of each function.
a. f ( x)  2 x 2  3x  1
up , up
b.
f ( x)  4 x 3  3x  1
down, up
c.
f ( x)  2 x  3x  1
down, down
d.
f ( x)  5x  3x  1
up, down
2
3
2
21. Find each thing listed and graph each function.
a.
f ( x) 
( x  4)( x  3)
( x  4)2 ( x 2  9)
Reduced Form (RF):
f (x ) 
YOU DO NOT NEED TO KNOW HOW TO
1
(x  4)(x  3)
GRAPH THIS ONE!!!!
Hole(s):
x=3
Vertical Asymptote (VA):
x = -4, x = -3
x and y intercepts:
x-int: none
y-int: (0,
1
)
12
end-behavior (top, bottom, even):
Horizontal Asymptote (HA):
y=0
b.
f ( x) 
x 2  3x
x2  x  6
9
8
7
6
5
4
3
2
1
Reduced Form (RF):
f (x ) 
x
x 2
Hole(s):
x = -3
Vertical Asymptote (VA):
x=2
x and y intercepts:
x-int: (0,0)
y-int: (0,0)
end-behavior (top, bottom, even):
Horizontal Asymptote (HA):
y=1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9