Factor x^3 + 1000

Step by step

The question

Question: factor x³ + 1000.

1 Write as cubes

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

x³ is x cubed.

1000 is 10 cubed.

So it is a sum of two cubes: a is x and b is 10.

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 = x and b = 10 into the pattern.

Careful!

It is not (x + 10)³.

Multiplied out, that has x² and x terms too.

3 Tidy up

x squared is x².

x times 10 is 10x.

10 squared is 100.

So we get (x + 10)(x² − 10x + 100).

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 x³ + 1000 again.

So it is right!

Let's look back.

First: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.