# 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} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}

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: x^{3}.

The middle sign in (x - 5)^{3} is minus.

So the signs of the right side terms alternate.

x^{3} is plus.

So write

-3 times,

square the first term, x^{2} times,

the last term, 5.

The signs of the right side terms alternate.

-3⋅x^{2}⋅5 is minus.

So write

+3 times,

the first term, x times,

square the last term, 5^{2}.

+3⋅x⋅5^{2} is plus.

So write minus, cube the last term, 5^{3}.

So (x - 5)^{3} = x^{3} - 3⋅x^{2}⋅5 + 3⋅x⋅5^{2} - 5^{3}.

-3⋅x^{2}⋅5 = -15x^{2}

+3⋅x⋅5^{2} = +75x

-5^{3} = -125

So x^{3} - 15x^{2} + 75x - 125 is the answer.