Difference of Two Cubes
There is a special case when multiplying polynomials that produces this: a3 - b3
Polynomials
A polynomial looks like this:
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| 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:
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 also get:
(a+b)(a2-ab+b2) = a3 + b3