52. 1. Factor 4 x 2 − 100 4 x squared , minus 100  by removing the common monomial factor and then using the difference-of-squares rule to factor the remaining expression.
    2. Factor 4 x 2 − 100 4 x squared , minus 100  by using the difference-of-squares rule and removing the common monomial factors.
    3. Reasoning Why can you factor 4 x 2 − 100 4 x squared , minus 100  in two different ways?
    4. Can you factor 3 x 2 − 75 3 x squared , minus 75  in the two ways you factored 4 x 2 − 100 4 x squared , minus 100  in parts (a) and (b)? Explain your answer.

C Challenge

Factor each expression.

53. 64 r 6 − 144 r 3 + 81 64 r to the sixth , minus . 144 r cubed . plus 81
54. p 6 + 40 p 3 q + 400 q 2 p to the sixth , plus , 40 p cubed , q plus . 400 q squared
55. 36 m 4 + 84 m 2 + 49 36 m to the fourth , plus , 84 m squared , plus 49
56. 108 n 6 − 147 108 n to the sixth . minus 147
57. x 20 − 4 x 10 y 5 + 4 y 10 x to the twentieth , minus , 4 x to the tenth . y to the fifth , plus , 4 y to the tenth
58. 256 g 4 − 100 h 6 256 g to the fourth . minus . 100 h to the sixth

59. The binomial 16 − 81 n 4 16 minus , 81 n to the fourth  can be factored twice using the difference-of-squares rule.

    1. Factor 16 − 81 n 4 16 minus , 81 n to the fourth  completely.
    2. Reasoning What characteristics do 16 and 81 n 4 81 n to the fourth  share that make this possible?
    3. Open-Ended Write another binomial that can be factored twice using the difference of squares rule.

Standardized Test Prep

SAT/ACT

60. What is the factored form of 4 x 2 − 20 x + 25 ? 4 x squared , minus 20 x plus 25 question mark

    1. ( 2 x + 5 ) ( 2 x − 5 ) open 2 x plus 5 close open 2 x minus 5 close
    2. ( 2 x − 5 ) ( 2 x − 5 ) open 2 x minus 5 close open 2 x minus 5 close
    3. ( 4 x − 5 ) ( 4 x − 5 ) open 4 x minus 5 close open 4 x minus 5 close
    4. ( 4 x + 5 ) ( 4 x − 5 ) open 4 x plus 5 close open 4 x minus 5 close

61. Which equation has − 2 negative 2  as its solution?

    6. x + 3 = 2 x + 1 x plus 3 equals 2 x plus 1
    7. x − 5 = 2 x − 7 x minus 5 equals 2 x minus 7
    8. 2 x + 5 = 5 x + 11 2 x plus 5 equals 5 x plus 11
    9. 3 x + 1 = x − 5 3 x plus 1 equals x minus 5

62. Which equation illustrates the Commutative Property of Multiplication?

    1. a b = b a eh b equals b eh
    2. a ( b c ) = ( a b ) c eh open b c close equals open eh b close c
    3. a b = a b eh b equals eh b
    4. a ( b + c ) = a b + a c eh open b plus c close equals eh b plus eh c

    Short Response

63. A film club sponsors a film fest at a local movie theater. Renting the theater costs $190. The admission is $2 per person.

    1. Write an equation that relates the film club's total cost c and the number of people p who attend the film fest.
    2. Graph the equation you wrote in part (a).

Mixed Review

See Lesson 8-6.

Factor each expression.

64. 18 x 2 + 9 x − 14 18 x squared , plus 9 x minus 14
65. 8 x 2 + 18 x + 9 8 x squared , plus 18 x plus 9
66. 12 x 2 − 41 x + 35 12 x squared , minus 41 x plus 35

Get Ready! To prepare for Lesson 8-8, do Exercises 67–69.

See Lesson 8-2.

Find the GCF of the terms of each polynomial.

67. 6 t 2 + 12 t − 4 6 t squared , plus 12 t minus 4
68. 9 m 3 + 15 m 2 − 21 m 9 m cubed , plus , 15 m squared , minus 21 m
69. 16 h 4 − 12 h 3 − 36 h 2 16 h to the fourth , minus , 12 h cubed , minus , 36 h squared

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