Associated Topics || Dr. Math Home || Search Dr. Math

### The Difference of Two Squares

```
Date: 9/21/95 at 20:28:48
From: Pat E Moshfeghian
Subject: Factoring

Dear Dr. Math,

In the following problem, if I factor the problem in Example A's way and then
factor the same problem with Example B's way, why is it that I do not get the

Example A:  x^6 - 64  (x to the power of 6 minus 64)
(x^3 - 8)(x^3 + 8)               factoring binomial
(x-2)(x^2+2x+4)(x+2)(x^2-2x+4)   factoring x^3 + y^3

Example B:   x^6 - 64
(x^2) ^3  - 4 ^3
(x^2 - 4) ( x^4 + 4x^2 + 16)
Factoring   x ^3 - y ^3
(x + 2 ) (x - 2 ) ( x ^ 4 + 4x ^2 + 16 )

after comparing the two answers we can conclude that

( x^4 + 4x ^ 2 + 16) = (x ^2 + 2x + 4 )( x ^2 - 2x + 4)

That means the left side of the equation is factorable.
My question is which law or rule are we using?

Thanks
```

```
Date: 9/21/95 at 21:10:41
From: Doctor Ken
Subject: Re: Factoring

Hello!

This is actually an example of the method "difference of two squares,"
the same method you used the first way you did the problem.

I'll show you why x^4 + 4x^2 + 16 = (x^2 + 2x + 4 )(x^2 - 2x + 4).

The easiest way is to multiply out the right side and see that it equals
the left side: write it as

[(x^2 + 4) + 2x][(x^2 + 4) - 2x]
(x^2 + 4)^2 - (2x)^2
(x^4 + 8x^2 + 16) - (4x^2)
x^4 + 4x^2 + 16, which is what we wanted.

If you had to factor x^4 + 4x^2 + 16, you could write it as

x^4 + 8x^2 + 16 - 4x^2
(x^2 + 4)^2 - (2x)^2
[x^2 + 4 - 2x][x^2 + 4 + 2x], as desired.

Rest assured, this is kind of a tricky problem.

-Doctor Ken, The Geometry Forum
```

```
Date: 9/21/95 at 21:35:46
From: Pat E Moshfeghian
Subject: Re: Factoring

Thank you Dr. Math,

The last time I mailed in a question it took about a day before I received
an answer, which in my book is good.  But wow! This time it was less than
an hour--This is GREAT!!!

Thank you,

Pat
```
Associated Topics:
High School Basic Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search