C Challenge

47. Use the Fundamental Theorem of Algebra and the Conjugate Root Theorem to show that any odd degree polynomial equation with real coefficients has at least one real root.
48. Reasoning What is the maximum number of points of intersection between the graphs of a quartic and a quintic polynomial function?
49. Reasoning What is the least possible degree of a polynomial with rational coefficients, leading coefficient 1, constant term 5, and zeros at 2 square root of 2  and 3 ? square root of 3 question mark  Show that such a polynomial has a rational zero and indicate this zero.

Standardized Test Prep

SAT/ACT

50. How many roots does f ( x ) = x 4 + 5 x 3 + 3 x 2 + 2 x + 6 f open x close equals , x to the fourth , plus 5 , x cubed , plus , 3 x squared , plus 2 x plus 6  have?

    1. 5
    2. 4
    3. 3
    4. 2
51. Which translation takes y = | x + 2 | − 1  to  y = | x | + 2 ? y equals vertical line x plus 2 vertical line negative 1 , to , y equals vertical line x vertical line plus 2 question mark
    6. 2 units right, 3 units down
    7. 2 units right, 3 units up
    8. 2 units left, 3 units up
    9. 2 units left, 3 units down
52. What is the factored form of the expression x 4 − 3 x 3 + 2 x 2 ? x to the fourth , minus , 3 x cubed , plus 2 , x squared , question mark
    1. x 2 ( x − 1 ) ( x + 2 ) x squared , open x minus 1 close open x plus 2 close
    2. x 2 ( x + 1 ) ( x + 2 ) x squared , open x plus 1 close open x plus 2 close
    3. x 2 ( x + 1 ) ( x − 2 ) x squared , open x plus 1 close open x minus 2 close
    4. x 2 ( x − 1 ) ( x − 2 ) x squared , open x minus 1 close open x minus 2 close

Short Response

53. How would you test whether ( 2 , − 2 ) open 2 comma negative 2 close  is a solution of the system? { y < − 2 x + 3 y ≥ x − 4 left brace . table with 2 rows and 1 column , row1 column 1 , y less than negative 2 x plus 3 , row2 column 1 , y greater than or equal to x minus 4 , end table

Mixed Review

See Lesson 5-5.

54. Find a fourth-degree polynomial equation with real coefficients that has 2 i 2 i  and − 3 + i negative 3 plus i  as roots.

See Lesson 4-7.

Solve each equation using the Quadratic Formula.

55. x 2 − 6 x + 1 = 0 x squared , minus 6 x plus 1 equals 0
56. 2 x 2 + 5 x = − 9 2 x squared , plus 5 x equals negative 9
57. 2 ( x 2 + 2 ) = 3 x 2 open , x squared , plus 2 close equals 3 x

See Lesson 4-3.

Determine whether a quadratic model exists for each set of values. If so, write the model.

58. f ( − 1 ) = 0 , f ( 2 ) = 3 , f ( 1 ) = 4 f open negative 1 close equals 0 comma f open 2 close equals 3 comma f open 1 close equals 4
59. f ( − 4 ) = 11 , f ( − 5 ) = 5 , f ( − 6 ) = 3 f open negative 4 close equals 11 comma f open negative 5 close equals 5 comma f open negative 6 close equals 3

Get Ready! To prepare for Lesson 5-7, do Exercises 60–65.

See Lesson 4-2.

Write each polynomial in standard form.

60. ( x + 1 ) 3 open x plus 1 close cubed
61. ( x − 3 ) 3 open x minus 3 close cubed
62. ( x − 2 ) 4 open x minus 2 close to the fourth
63. ( x − 1 ) 2 open x minus 1 close squared
64. ( x + 5 ) 3 open x plus 5 close cubed
65. ( 4 − x ) 3 open 4 minus x close cubed

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