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### 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 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/
```
Associated Topics:
High School Polynomials
Middle School Factoring Expressions

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