45. Gardening A gardener is designing a new garden in the shape of a trapezoid. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If she has enough topsoil to create a 60  ft 2 60 , ft squared  garden, what dimensions should she use for the garden?
46. Open-Ended Write a fourth-degree polynomial equation with integer coefficients that has two irrational roots and two imaginary roots.

C Challenge

47. 1. Find a polynomial equation in which 1 + 2 1 plus square root of 2  is the only root.
    2. Find a polynomial equation with root 1 + 2 1 plus square root of 2  of multiplicity 2.
    3. Find c such that 1 + 2 1 plus square root of 2  is a root of x 2 − 2 x + c = 0 . x squared , minus 2 x plus c equals 0 .
48. 1. Using real and imaginary as types of roots, list all possible combinations of root type for a fourth-degree polynomial equation.
    2. Repeat the process for a fifth-degree polynomial equation.
    3. Make a Conjecture Make a conjecture about the number of real roots of an odd-degree polynomial equation.
49. Writing A student states that 2 + 3 2 plus square root of 3  is a root of x 2 − 2 x − ( 3 + 2 3 ) = 0 . x squared , minus 2 x minus . open . 3 plus 2 square root of 3 . close . equals 0 . .  The student claims that 2 − 3 2 minus square root of 3  is another root of the equation by the Conjugate Root Theorem. Explain how you would respond to the student.

Standardized Test Prep

GRIDDED RESPONSE

SAT/ACT

50. What is a positive root of − 5 x 3 − 2 x 2 + 9 x + 30 = 0 ? negative 5 , x cubed , minus , 2 x squared , plus 9 x plus 30 equals 0 question mark
51. What is the remainder when you divide x 3 + 2 x 2 − x − 6 x cubed , plus 2 , x squared , minus x minus 6  by x − 1 ? x minus 1 question mark
52. A polynomial with rational coefficients has roots − 3 i negative 3 i  and 8 + 7 . 8 plus square root of 7 .  What is the minimum degree of the polynomial?

53. What is the value of y in the solution of the system of equations?

    { 10 x + 24 y = 9 8 x + 60 y = 14 left brace . table with 2 rows and 2 columns , row1 column 1 , 10 x plus 24 y , column 2 equals 9 , row2 column 1 , 8 x plus 60 y , column 2 equals 14 , end table

54. What is the value of the greater solution of the equation 6 x 2 − 17 x + 5 = 0 ? 6 x squared , minus 17 x plus 5 equals 0 question mark

Mixed Review

See Lesson 5-4.

Divide.

55. ( x 3 + 5 x 2 − 3 ) ÷ ( x − 1 ) open , x cubed , plus , 5 x squared , minus 3 close divides open x minus 1 close
56. ( 8 x 3 + 12 x 2 + 7 ) ÷ ( x + 6 ) open 8 , x cubed , plus , 12 x squared , plus 7 close divides open x plus 6 close
57. ( 7 x 2 + 11 x − 4 ) ÷ ( x + 2 ) open 7 , x squared , plus 11 x minus 4 close divides open x plus 2 close

See Lesson 4-8.

Solve.

58. 7 x 2 + 63 = 0 7 x squared , plus 63 equals 0
59. x 2 + 81 = 0 x squared , plus 81 equals 0
60. 2 x 2 + 288 = 0 2 x squared , plus 288 equals 0

Get Ready! To prepare for Lesson 5-6, do Exercises 61 and 62.

See Lesson 5-1.

Write each polynomial in standard form. Then classify it by degree and by number of terms.

61. 6 x 2 + 11 − 5 x 4 + 9 x 6 x squared , plus 11 minus , 5 x to the fourth , plus 9 x
62. 13 x − 4 x 5 + 7 x 3 + 2 13 x minus , 4 x to the fifth , plus , 7 x cubed , plus 2

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