Difference of Two Cubes

There is a special case when multiplying polynomials that produces this: a3 - b3

Polynomials

A polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

Difference of Two Cubes

This is a special case when multiplying polynomials called the Difference of Two Cubes:

(a-b)(a2+ab+b2) = a3 - b3

See why it works out so simply:

Example from Geometry:

Take two cubes of lengths x and y:

polynomial cubes

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

polynomial cubes

The volumes of these boxes are:

  • A = y3
  • B = x2(x – y)
  • C = xy(x – y)
  • D = y2(x – y)

But together, A, B, C and D make up the larger cube that has volume x3:

x3  =  y3 + x2(x – y) + xy(x – y) + y2(x – y)
x3 – y3  =  x2(x – y) + xy(x – y) + y2(x – y)
x3 – y3  =  (x – y)(x2 + xy + y2)

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 also get:

(a+b)(a2-ab+b2) = a3 + b3