Factor 8 - x^3

Step by step

The question

Question: factor −x³ + 8.

1 Common factor first

Always look for a common factor first.

The first term is negative, so we take out the minus.

So it is −(x³ − 8).

2 Write as cubes

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

x³ is x cubed.

8 is 2 cubed.

So it is a difference of two cubes: a is x and b is 2.

3 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 = 2 into the pattern.

Careful!

It is not (x − 2)³.

Multiplied out, that has x² and x terms too.

4 Tidy up

x squared is x².

x times 2 is 2x.

2 squared is 4.

So we get −(x − 2)(x² + 2x + 4).

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

5 Check

Let's check: we multiply it back out.

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

Then the common factor times every part.

We get −x³ + 8 again.

So it is right!

Let's look back.

First: common factor first.

Then: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.