Step by step
The question
Question: factor 125x³ − 1.
1 Write as cubes
Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.
125x³ is 5x cubed.
1 is 1 cubed: 1 × 1 × 1 = 1.
So it is a difference of two cubes: a is 5x 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 = 5x and b = 1 into the pattern.
Careful!
It is not (5x − 1)³.
Multiplied out, that has x² and x terms too.
3 Tidy up
5x squared is 25x².
5x times 1 is 5x.
1 squared is 1.
So we get (5x − 1)(25x² + 5x + 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 125x³ − 1 again.
So it is right!
Let's look back.
First: write as cubes.
Then: use the pattern.
Then: tidy up.
Last: check.