Special Products and Factoring Strategies
Review of Three Special Products
Recall the three special products:

Difference of Squares
x^{2}  y^{2} = (x  y) (x + y)

Square of Sum
x^{2} + 2xy + y^{2} = (x + y)^{2}

Square of Difference
x^{2}  2xy + y^{2} = (x  y)^{2}
^{
}
Special Products Involving Cubes
Just as there is a difference of squares formula, there is also a difference of
cubes formula.

x^{3}  y^{3} = (x  y) (x^{2} + xy +
y^{2})
Proof:
We use the distributive law on the right hand side
x (x^{2} + xy + y^{2}) 
y (x^{2} + xy + y^{2})
= x^{3} + x^{2}y + xy^{2}  x^{2}y 
xy^{2}  y^{3}

Now combine like terms to get
x^{3}  y^{3
}
Next, we state the sum of cubes formula.^{
}

x^{3} + y^{3}
= (x + y)(x^{2}  xy
+ y^{2})
Exercise
Prove the sum of cubes equation
(Equation 5)
Using the Special Product Formulas for Factoring
Examples:
Factor the following

36x^{2}  4y^{2} = (6x 
2y) (6x + 2y)
Notice that there only two terms.
3x^{3}  12x^{2} + 12x
= 3x (x^{2}  4x +
4) Remember to pull the GCF out first.
= 3x(x 2)^{2}
x^{6}  64 =
(x^{3 } 8) (x^{3} + 8)
= (x  2) (x^{2} + 2x + 4) (x + 2) (x^{2
} 2x + 4)
Exercises:
Factor the following

45a^{3}b  20ab^{3}

64x^{6}  16x^{3} + 1

x^{2} + 2xy + y^{2}  81

x^{12}  y^{12}
(Challenge Problem)
Factoring Strategies

Always pull out the GCF first

Look for special products. If there are only two terms then
look for sum of cubes or difference of squares or cubes. If there are
three terms, look for squares of a difference or a sum.

If there are three terms and the first coefficient is 1 then use
simple trinomial factoring.

If there are three terms and the first coefficient is not 1 then
use the AC method.

If there are four terms then try factoring by grouping.
Exercises

x^{3 } x

x^{2 } 7x  30

96a^{2 }b  48ab  72a + 36

4x^{2 } 36xy + 81y^{2 }

5a^{4}b^{3 }+ 1080a

2x^{2 } + 5x  12

5x^{3 }+ 40

x^{3 }+ 3x^{2 } 4x  12
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