C Challenge

Use the Binomial Theorem to expand each complex expression.

47. ( 7 + − 16 ) 5 open . 7 plus , square root of negative 16 end root . close to the fifth
48. ( − 81 − 3 ) 3 open . square root of negative 81 end root , minus 3 . close cubed
49. ( x 2 − i ) 7 open , x squared , minus i close to the seventh
50. The first term in the expansion of a binomial ( a x + b y ) n open eh x plus b y close to the n  is 1024 x 10 . x to the tenth , .  Find a and n.
51. Determine the coefficient of x 7 y x to the seventh , y  in the expansion of ( 1 2 x + 1 4 y ) 8 . open . 1 half , x plus , 1 fourth , y . close to the eighth . .
52. 1. Expand ( 1 + i ) 4 . open 1 plus i close to the fourth . .
    2. Verify that 1 − i 1 minus i  is a fourth root of − 4 negative 4  by repeating the process in part (a) for ( 1 − i ) 4 . open 1 minus i close to the fourth . .
53. Verify that − 1 + 3 i negative 1 plus square root of 3 i  is a cube root of 8 by expanding ( − 1 + 3 i ) 3 . open . negative 1 plus square root of 3 i . close cubed . .

Standardized Test Prep

SAT/ACT

54. What is the fourth term in the expansion of ( 2 a + 4 b ) 5 ? open 2 eh plus 4 b close to the fifth . question mark
    1. 256 a 4 b 256 , eh to the fourth , b
    2. 768 a 3 b 2 768 , eh cubed , b squared
    3. 2560 a 2 b 3 2560 , eh squared , b cubed
    4. 2048 a b 4 2048 , eh , b to the fourth

55. Suppose y varies directly with x. If x is 30 when y is 10, what is x when y is 9?

    6. 3
    7. 27
    8. 29
    9. 300 9 300 over 9
56. Which of following is a root of 9 x 2 − 30 x + 25 = 0 ? 9 x squared , minus 30 x plus 25 equals 0 question mark
    1. x = 3 5 x equals , 3 fifths
    2. x = 5 3 x equals , 5 thirds
    3. x = − 5 3 x equals negative , 5 thirds
    4. x = − 3 5 x equals negative , 3 fifths

Extended Response

57. One company charges a monthly fee of $7.95 and $2.25 per hour for Internet access. Another company does not charge a monthly fee, but charges $2.75 per hour for Internet access. Write a system of equations to represent the cost c for t hours of access in one month for each company. Then find how many hours of use it will take for the costs to be equal.

Mixed Review

See Lesson 5-6.

Find all the roots of each equation.

58. x 4 + 7 x 3 + 20 x 2 + 29 x + 15 = 0 x to the fourth , plus 7 , x cubed , plus , 20 x squared , plus 29 x plus 15 equals 0
59. x 5 − x 4 + 10 x 3 − 10 x 2 + 9 x − 9 = 0 x to the fifth , minus , x to the fourth , plus 10 , x cubed , minus , 10 x squared , plus 9 x minus 9 equals 0
60. 2 x 3 + 11 x 2 + 14 x + 8 = 0 2 x cubed , plus 11 , x squared , plus 14 x plus 8 equals 0
61. x 4 − x 3 + 6 x 2 − 13 x + 7 = 0 x to the fourth , minus , x cubed , plus 6 , x squared , minus 13 x plus 7 equals 0

See Lesson 4-8.

Simplify each expression.

62. ( 5 i − 4 ) ( − 2 i + 7 ) open 5 i minus 4 close open negative 2 i plus 7 close
63. ( − 3 i ) ( 20 i ) ( 10 i ) open negative 3 i close open 20 i close open 10 i close
64. − 6 − 2 i 3 + i fraction negative 6 minus 2 i , over 3 plus i end fraction
65. 11 i + 9 2 − i fraction 11 i plus 9 , over 2 minus i end fraction

Get Ready! To prepare for Lesson 5-8, do Exercises 66–68.

See Lesson 5-1.

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

66. 5 x 2 − x + 2 x 3 + 9 5 x squared , minus x plus , 2 x cubed , plus 9
67. 1 + 4 x − 7 x 2 1 plus 4 x minus , 7 x squared
68. − 9 x 2 + x − 3 x 3 − 8 + 12 x 4 negative 9 , x squared , plus x minus , 3 x cubed , minus 8 plus , 12 x to the fourth

Table of Contents