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Name: _______________________ Date: ___________ Period: __________ 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 x4 b. 2 x x 20 6 2 c. x x4 1 6 d. 2 x4 ( 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) x4 ( 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