Algebra I in Simple English/Factoring/Factoring a^2-b^2 Binomials

Difference of Squares

Any binomial of the form $a^2-b^2$ may be written as $(a+b)\cdot (a-b)$. That is

$a^2-b^2=(a-b)\cdot (a+b)$.

Example 1: Factor $x^2-9$.

This is clearly seen just take $a^2=x^2$ and $b^2=9$ so that $b=3$. So $x^2-9=(x-3)(x+3)$

Example 2:: $32w^4-162$.

Here is is unclear where we can use the difference of squares as 32 is NOT a perfect square. However if we look we see that we can factor out a common factor of 2.

$32w^4-162=2(16w^4-81)$

Now we see we can use the difference of two squares to simplify matters take $a^2=16w^4$ and $b^2=81$:

$2(16w^4-81)=2(4w^2-9)(4w^2+9)$

Now we notice that we can use the difference of squares again in the first factor to get:

$2(4w^2-9)(4w^2+9)=2(2w+3)(2w-3)(4w^2+9)$

This is now completely factored.

This is brings us to our next point that is that $a^2+b^2$ is NOT FACTORABLE (at least for the purposes of this class).