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Factoring
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Factoring Special Products

Differences of Two Squares
The factors of a difference of two squares are the sum and difference of the expressions being squared; that is,

x 2 − y 2 =(x+y)(x−y).

Example

Factor each polynomial completely.

1. a 2 −4
2. 4 a 2 −9 b 2
3. 16 b 2 c 2 −1
4. 64a x 2 −49a y 2
5. x 4 − y 4
6. (a+b) 2 − (c+d) 2

Solution
1. a 2 −4=( a+2 )( a−2 )

2. 4 a 2 −9 b 2
= ( 2a ) 2 − ( 3b ) 2
=( 2a+3b )( 2a−3b )

3. 16 b 2 c 2 −1
= ( 4bc ) 2 − 1 2
=( 4bc+1 )( 4bc−1 )

4. 64a x 2 −49a y 2
=a( 64 x 2 −49 y 2 )
=a[ ( 8x ) 2 − ( 7y ) 2 ]
=a( 8x+7y )( 8x−7y )

5. x 4 − y 4
= ( x 2 ) 2 − ( y 2 ) 2
=( x 2 + y 2 )( x 2 − y 2 )
=( x 2 + y 2 )( x+y )( x−y )

6. (a+b) 2 − (c+d) 2
=[ ( a+b )+( c+d ) ]
[ ( a+b )−( c+d ) ]
=( a+b+c+d )( a+b−c−d )