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

'''Difference of Squares'''

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

<math>a^2 - b^2 = (a-b) \cdot (a+b) </math>.

'''Example 1:''' Factor <math>x^2 - 9</math>. 

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


'''Example 2:''': <math>32w^4 - 162</math>.  

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. 

<math>32w^4 - 162 = 2(16w^4 -81)</math> 

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

<math>2(16w^4 -81) =2(4w^2 - 9)(4w^2 + 9) </math>

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

<math>2(4w^2 - 9)(4w^2 + 9)=2(2w+3)(2w-3)(4w^2 + 9)</math>

This is now completely factored.


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