Difference of Two Cubes
A special case when multiplying polynomials that produces this: a3 − b3
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)(a2+ab+b2) = a3 − b3
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 = 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 get:
(a+b)(a2−ab+b2) = a3 + b3
(note the minus in front of "ab" also)