Difference between Two Cubes

Date: 11/09/2001 at 14:13:06
From: Robert Sullivan
Subject: Difference between two cubes

To Dr.Math,

Many people know the equation:

a^2 - b^2 = (a+b) * (a-b)

But what is another equation for

a^3 - b^3 

I know that a^3 - b^3 = (a*a*a)-(b*b*b), but I'm not an expert on 
cancelling and transforming equations. Is it possible?

Robert Sullivan

Date: 11/09/2001 at 15:10:50
From: Doctor Peterson
Subject: Re: Difference between two cubes

Hi, Robert.

Yes, you can factor a^3 - b^3. I'll show you a way to do this.

Looking at a^2 - b^2, you can see that it will be zero if a = b. If 
you have seen anything about solving quadratic equations by factoring, 
you will recognize that this means that a^2 - b^2 can be factored, 
with one of the factors being (a-b), which is zero when a = b.

The same thing happens with a^3 - b^3; this is also zero when a = b, 
so it can be factored as (a-b) times something. You can find the other 
factor by dividing:

          ________a^2___+_ab____+_b^2_
    a - b ) a^3                 - b^3
            a^3 - a^2 b
            -----------
                  a^2 b
                  a^b b - a b^2
                  -------------
                          a b^2 - b^3
                          a b^2 - b^3
                          -----------
                                    0

(This is long division of polynomials, which works much like long 
division of numbers.)

This tells us that

    a^3 - b^3 = (a - b)(a^2 + ab + b^2)

I think that is what you were looking for.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/