Factor a^3 + 1

Step by step

The question

Question: factor a³ + 1.

1 Write as cubes

Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.

a³ is a cubed.

1 is 1 cubed: 1 × 1 × 1 = 1.

So it is a sum of two cubes: a is a and b is 1.

2 Use the pattern

Here is the pattern.

The first bracket has the same sign as the question.

In the second bracket the middle sign is the opposite one, and the last sign is always plus.

We put a = a and b = 1 into the pattern.

Careful!

It is not (a + 1)³.

Multiplied out, that has a² and a terms too.

3 Tidy up

a squared is a².

a times 1 is a.

1 squared is 1.

So we get (a + 1)(a² − a + 1).

The second bracket can't be factored: with this pattern it never can.

4 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

We get a³ + 1 again.

So it is right!

Let's look back.

First: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.