8-7 Factoring Special Cases

Quick Review

When you factor a perfect-square trinomial, the two binomial factors are the same.

a 2 + 2 a b + b 2 = ( a + b ) ( a + b ) = ( a + b ) 2 a 2 − 2 a b + b 2 = ( a − b ) ( a − b ) = ( a − b ) 2 table with 2 rows and 1 column , row1 column 1 , eh squared , plus 2 eh b plus , b squared , equals open eh plus b close open eh plus b close equals . open eh plus b close squared , row2 column 1 , eh squared , minus 2 eh b plus , b squared , equals open eh minus b close open eh minus b close equals . open eh minus b close squared , end table

When you factor a difference of squares of two terms, the two binomial factors are the sum and the difference of the two terms.

a 2 − b 2 = ( a + b ) ( a − b ) eh squared , minus , b squared , equals open eh plus b close open eh minus b close

Example

What is the factored form of 81 t 2 − 90 t + 25 ? 81 t squared , minus 90 t plus 25 question mark

First rewrite the first and last terms as squares. Then determine if the middle term equals − 2 a b . negative 2 eh b .

81 t 2 − 90 t + 25 =   ( 9 t ) 2 − 90 t + 5 2   = ( 9 t ) 2 − 2 ( 9 t ) ( 5 ) + 5 2   = ( 9 t − 5 ) 2 table with 3 rows and 2 columns , row1 column 1 , 81 t squared , minus 90 t plus 25 , column 2 equals . open , 9 t , close squared . minus 90 t plus , 5 squared , row2 column 1 , , column 2 equals . open 9 t close squared . minus 2 open 9 t close open 5 close plus , 5 squared , row3 column 1 , , column 2 equals . open 9 t minus 5 close squared , end table

Exercises

Factor each expression.

59. s 2 − 20 s + 100 s squared , minus 20 s plus 100
60. 16 q 2 + 56 q + 49 16 q squared , plus 56 q plus 49
61. r 2 − 64 r squared , minus 64
62. 9 z 2 − 16 9 z squared , minus 16
63. 25 m 2 + 80 m + 64 25 m squared , plus 80 m plus 64
64. 49 n 2 − 4 49 n squared , minus 4
65. g 2 − 225 g squared , minus 225
66. 9 p 2 − 42 p + 49 9 p squared , minus 42 p plus 49
67. 36 h 2 − 12 h + 1 36 h squared , minus 12 h plus 1
68. w 2 + 24 w + 144 w squared , plus 24 w plus 144
69. 32 v 2 − 8 32 v squared , minus 8
70. 25 x 2 − 36 25 x squared , minus 36
71. Geometry Find an expression for the length of a side of a square with an area of 9 n 2 + 54 n + 81 . 9 n squared , plus 54 n plus 81 .
72. Reasoning Suppose you are using algebra tiles to factor a quadratic trinomial. What do you know about the factors of the trinomial when the tiles form a square?

8-8 Factoring by Grouping

Quick Review

When a polynomial has four or more terms, you may be able to group the terms and find a common binomial factor. Then you can use the Distributive Property to factor the polynomial.

Example

What is the factored form of 2 r 3 − 12 r 2 + 5 r − 30 ? 2 r cubed , minus , 12 r squared , plus 5 r minus 30 question mark

First factor out the GCF from each group of two terms. Then factor out a common binomial factor.

2 r 3 − 12 r 2 + 5 r − 30 = 2 r 2 ( r − 6 ) + 5 ( r − 6 )   = ( 2 r 2 + 5 ) ( r − 6 ) table with 2 rows and 2 columns , row1 column 1 , 2 r cubed , minus , 12 r squared , plus 5 r minus 30 , column 2 equals , 2 r squared , open r minus 6 close plus 5 open r minus 6 close , row2 column 1 , , column 2 equals open , 2 r squared , plus 5 close open r minus 6 close , end table

Exercises

Find the GCF of the first two terms and the GCF of the last two terms for each polynomial.

73. 6 y 3 − 3 y 2 + 2 y − 1 6 y cubed , minus , 3 y squared , plus 2 y minus 1
74. 8 m 3 + 40 m 2 + 6 m + 15 8 m cubed , plus , 40 m squared , plus 6 m plus 15

Factor completely.

75. 6 d 4 + 4 d 3 − 6 d 2 − 4 d 6 d to the fourth , plus , 4 d cubed , minus , 6 d squared , minus 4 d
76. 11 b 3 − 6 b 2 + 11 b − 6 11 b cubed , minus , 6 b squared , plus 11 b minus 6
77. 45 z 3 + 20 z 2 + 9 z + 4 45 z cubed , plus , 20 z squared , plus 9 z plus 4
78. 9 a 3 − 12 a 2 + 18 a − 24 9 eh cubed , minus , 12 eh squared , plus 18 eh minus 24

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