# Difference of Two Cubes

*A special case when multiplying polynomials that produces this: a^{3} − b*

^{3}## Polynomials

A polynomial looks like this:

example of a polynomial

## Difference of Two Cubes

The **Difference of Two Cubes** is a special case of multiplying polynomials that looks like this:

(a−b)(a^{2}+ab+b^{2}) = a^{3} − b^{3}

It comes up sometimes when doing solutions, so is worth remembering.

And this is why it works out (press play):

## Example from Geometry:

Take two cubes of lengths x and y:

The larger "x" cube can be split into four smaller boxes (cuboids), with box **A being a cube of size "y"**:

The volumes of these boxes are:

- A = y
^{3} - B = x
^{2}(x − y) - C = xy(x − y)
- D = y
^{2}(x − y)

But together, A, B, C and D make up the larger cube that has volume x^{3}:

x^{3} |
= | y^{3} + x^{2}(x − y) + xy(x − y) + y^{2}(x − y) |

x^{3} − y^{3} |
= | x^{2}(x − y) + xy(x − y) + y^{2}(x − y) |

x^{3} − y^{3} |
= | (x − y)(x^{2} + xy + y^{2}) |

Hey! We ended up with the same formula! Thank goodness.

## Sum of Two Cubes

There is also the "Sum of Two Cubes"

By changing the sign of b in each case we get:

(a+b)(a^{2}−ab+b^{2}) = a^{3} + b^{3}

(note the minus in front of "ab" also)