Cube of a Difference (a - b)3

How to solve the cube of a difference (a - b)3: formula, 1 example, and its solution.

Formula

Formula

(a + b)3 = a3 - 3a2b + 3ab2 - b3

The middle sign in (a - b)3 is minus.
So the signs of the right side terms alternate: plus, minus, plus, minus.
This is the difference between
the (a - b)3 formula and the (a + b)3 formula.

Example

Example

Solution

First cube the first term: x3.

The middle sign in (x - 5)3 is minus.
So the signs of the right side terms alternate.
x3 is plus.

So write
-3 times,
square the first term, x2 times,
the last term, 5.

The signs of the right side terms alternate.
-3⋅x2⋅5 is minus.

So write
+3 times,
the first term, x times,
square the last term, 52.

+3⋅x⋅52 is plus.

So write minus, cube the last term, 53.

So (x - 5)3 = x3 - 3⋅x2⋅5 + 3⋅x⋅52 - 53.

-3⋅x2⋅5 = -15x2
+3⋅x⋅52 = +75x
-53 = -125

So x3 - 15x2 + 75x - 125 is the answer.