6.4 Factoring Special Forms
Objectives:
1. Factor difference of squares
2. Factor perfect square trinomials
3. Factor sum and difference of two cubes
When discussing products in the last chapter, we looked at:
The product of the sum and difference of two terms:
( x + y )( x − y ) = _________________
We see that this product has the form: ________________________
Factoring the Difference of Two Squares
This gives us a way to factor any expression that is the difference of two
squares.
x 2 − y 2 = ( x + y )( x − y )
To do this, it is necessary that you can notice perfect squares quickly:
EXAMPLE: List the perfect squares of 1 through 15:
1, 4, ...
We see that the formula to factor the difference of two squares requires us to
notice what terms are being squared.
2
2
16a 2 − 81b 2 = ( 4a ) − ( 9b )
121x 4 − 49 y 8 = (11x 2 ) − ( 7 y 4 )
2
EXAMPLE: Write the following in the form:
2
( __ ) − ( __ )
2
2
a.) 225 x 2 − 4 y 2
b.) 196 x10 − 25 y16
1
To factor the difference of two squares: 16a 2 − 81b 2 = ( 4a ) − ( 9b )
1.) Find the terms that are squared: 4a & 9b
2.) Take the sum of the two terms times the difference of the two
terms
2
2
16a 2 − 81b 2 = ( 4a ) − ( 9b ) = ( 4a + 9b )( 4a − 9b )
2
2
We can’t factor the sum of two squares. It is called a prime polynomial.
EXAMPLE: Factor
a.) 4 x 2 − 81
b.) 9 x 2 + 121
c.) 9m 2 − 64n 4
e.) 225 x 2 − 4 y 2
2
f.) 196 x10 − 25 y16
When factoring any expression, we must always factor out the
__________________ first.
EXAMPLE: Factor the following
a.) 6 p 2 q 2 s 2 − 54r 2 s 2
b.) 81x 3 − 49 x
c.) x 4 − 16
d.) 3a 3 − 12a + 3a 2b − 12b
e.) ( 3 x − 1) − 49
2
3
f.) 48a 5 − 3ab 4
Factoring Perfect Square Trinomials:
Notice the form that a perfect square trinomial takes on.
2
( x + y ) = x 2 + 2 xy + y 2
( x − y)
2
= x 2 − 2 xy + y 2
EXAMPLE: Factor
a.) 4 x 2 − 20 xy + 25 y 2
b.) a 2 + 6ab + 9b 2
EXAMPLE: Find the value of k so that the polynomial is a perfect square
trinomial
4 x 2 + 12 x + k
Factoring The Sum or Difference of Two Cubes:
We also have nice formulas for the sum and difference of two cubes:
a 3 − b3 = ( a − b ) ( a 2 + ab + b 2 )
a 3 + b3 = ( a + b ) ( a 2 − ab + b 2 )
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To do this, it is necessary that you can notice perfect cubes quickly:
EXAMPLE: List the perfect cubes of 1 through 7:
1, 8, ___, ____, 125, ___, ____
Factor 125 x 3 − 216
125 x3 − 216 = ( 5 x ) − ( 6 ) = ( a − b ) ( a 2 + ab + b 2 )
3
3
(
= ( 5 x − 6 ) ( 5 x ) + ( 5 x )( 6 ) + ( 6 )
2
= ( 5 x − 6 ) ( 25 x 2 + 30 x + 36 )
2
)
EXAMPLE: Factor each of the following:
a.) 125 x 3 + 8
b.) 64 y − y 4
c.) 8 x3 − 27 y 3
5