Expand (x - 1)^3

Step by step

The question

Question: expand (x − 1)³.

Careful!

A power of a bracket is not the power of each part.

(x − 1) to the power 3 is the bracket times itself 3 times.

There are middle terms too.

1 Use Pascal's triangle

Pascal's triangle starts with 1.

Every row starts and ends with 1.

Each other number is the two numbers above it added.

2 The pattern

There are 4 terms.

The numbers in front come from row 3: 1, 3, 3, 1.

The powers of x go down from 3 to 0.

The powers of −1 go up from 0 to 3.

Remember: any number to the power 0 is 1.

3 Work out each term

The first term: 1, x to the power 3, −1 to the power 0.

That is x³.

Second term: 3 times −1 is −3.

So −3x².

Third term: (−1) to the power 2 is 1.

So 3x.

Fourth term: (−1) to the power 3 is −1.

So −1.

4 Add them up

So (x − 1)³ is x³ − 3x² + 3x − 1.

5 Check

Let's check with x = 2.

The question: 1 to the power 3 is 1.

Our answer with x = 2: the terms add up to 1 too.

It works!

Let's look back.

First: use Pascal's triangle.

Then: the pattern.

Then: work out each term.

Then: add them up.

Last: check.