# Factor Polynomials Using Special Products

## Key Questions

You identify special products by their values if its a perfect square or cubes..

#### Explanation:

For examples;

Difference of two squares is: ${x}^{2} - {y}^{2} = \left(x + y\right) \left(x - y\right)$

${\left(x + y\right)}^{2} = {x}^{2} + {y}^{2} + 2 a b$

${\left(x - y\right)}^{2} = {x}^{2} + {y}^{2} - 2 a b$

Note: ${\left(x - y\right)}^{2} \ne {x}^{2} - {y}^{2}$

• There's a single formula which refers to "difference of squares":

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

If we use FOIL we can prove that. Difference of squares method would refer to doing something like the following:

${x}^{2} - 1 = \left(x - 1\right) \left(x + 1\right)$
${x}^{2} - 4 = \left(x - 2\right) \left(x + 2\right)$

Or even the double application here
${x}^{4} - 16 = {\left({x}^{2}\right)}^{2} - {4}^{2} = \left({x}^{2} - 4\right) \left({x}^{2} + 4\right) = \left(x - 2\right) \left(x + 2\right) \left({x}^{2} + 4\right)$

• Look for numbers that are perfect squares or perfect cubes.
There are many special products in factoring. Three of the most well-known are
${\left(x + y\right)}^{2} = {x}^{2} + 2 x y + {y}^{2}$
and
(x−y)^2=x-2xy+y^2
and
(x+y)(x−y)=x^2−y^2

Two less well-known ones are
x^3+y^3=(x+y)(x^2−xy+y^2)
and
x^3−y^3=(x−y)(x^2+xy−y^2)
Note that in an actual problem, x and y can be any number or variable. Hope this helped!