SECTION 4.4 Complex Zeros; Fundamental Theorem of Algebra 235 In Problems 25–32, use the given zero to find the remaining zeros of each polynomial function. 25. f x x x x 5 9 45; 3 2 ( ) = − + − zero: i3 26. g x x x x 3 25 75; 3 2 ( ) = + + + zero: i5− 27. ( ) = + + + − f x x x x x 4 7 62 112 32; 4 3 2 zero: i4− 28. ( ) = + + + − h x x x x x 3 5 25 45 18; 4 3 2 zero: i3 29. h x x x x x 7 23 15 522; 4 3 2 ( ) = − + − − zero: i 2 5 − 30. f x x x x x 7 14 38 60; 4 3 2 ( ) = − + − − zero: i 1 3 + 31. h x x x x x x 3 2 9 6 84 56; 5 4 3 2 ( ) = + − − − − zero: i2− 32. g x x x x x x 2 3 5 15 207 108; 5 4 3 2 ( ) = − − − − + zero: i3 In Problems 33–42, find the complex zeros of each polynomial function. Write f in factored form. 33. f x x 1 3 ( ) = − 34. f x x 1 4 ( ) = − 35. f x x x x 8 25 26 3 2 ( ) = − + − 36. f x x x x 13 57 85 3 2 ( ) = + + + 37. f x x x5 4 4 2 ( ) = + + 38. f x x x 13 36 4 2 ( ) = + + 39. f x x x x x 2 22 50 75 4 3 2 ( ) = + + + − 40. f x x x x x 3 19 27 252 4 3 2 ( ) = + − + − 41. f x x x x x 3 9 159 52 4 3 2 ( ) = − − + − 42. f x x x x x 2 35 113 65 4 3 2 ( ) = + − − + 43. Challenge Problem Suppose f x x x bx 2 14 3 3 2 ( ) = − + − with f 2 0 ( ) = and g x x cx x8 30, 3 2 ( ) = + − + with the zero x i 3 , = − where b and c are real numbers. Find f g 1 . ( )( ) ⋅ † 44. Challenge Problem Let f be the polynomial function of degree 4 with real coefficients, leading coefficient 1, and zeros x i 3 , 2, 2. = + − Let g be the polynomial function of degree 4 with intercept 0, 4 ( ) − and zeros x i i , 2. = Find f g 1 . ( )( ) + † 45. Challenge Problem The complex zeros of ( ) = + f x x 1 4 For the function f x x 1: 4 ( ) = + (a) Factor f into the product of two irreducible quadratics. (b) Find the zeros of f by finding the zeros of each irreducible quadratic. 46. f is a polynomial function of degree 3 whose coefficients are real numbers; its zeros are 2, i, and i 3 . + 47. f is a polynomial function of degree 3 whose coefficients are real numbers; its zeros are i i 4 , 4 , + − and i 2 . + 48. f is a polynomial function of degree 4 whose coefficients are real numbers; two of its zeros are 3− and i 4 . − Explain why one of the remaining zeros must be a real number.Write down one of the missing zeros. 49. f is a polynomial function of degree 4 whose coefficients are real numbers; three of its zeros are 2, i 1 2 , + and i 1 2 . − Explain why the remaining zero must be a real number. 50. For the polynomial function ( ) = + − f x x ix 2 10: 2 (a) Verify that i 3 − is a zero of f. (b) Verify that i 3 + is not a zero of f. (c) Explain why these results do not contradict the Conjugate Pairs Theorem. Explaining Concepts In Problems 46 and 47, explain why the facts given are contradictory. ‘Are You Prepared?’ Answers 1. Sum: i3 ; product: + i 1 21 2. − − − + i i 1 , 1 3. − − i 3 4 4. 29 Retain Your Knowledge Problems 51–60 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 51. Draw a scatter plot for the given data. x 1− 1 2 5 8 10 y 4− 0 3 1 5 7 52. Given f x x 3 , ( ) = − find x so that f x 5. ( ) = 53. Multiply: x x x 2 5 3 4 2 ( ) ( ) − + − 54. Find the area and circumference of a circle with a diameter of 6 feet. 55. If f x x x 1 ( ) = + and g x x3 2, ( ) = − find g f x( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ and state its domain. 56. Solve x y 3 5 = + − for y. 57. Find the domain of g x x x x 2 . ( ) = − + 58. Find the intercepts of the graph of the equation x y 3 12. 2 + = 59. Rationalize the numerator: x x 3 7 − + 60. Find the difference quotient of ( ) = + f x x 8. 3 † Courtesy of the Joliet Junior College Mathematics Department
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