Step by step
The question
Question: factor 24x³ + 81.
1 Common factor first
Always look for a common factor first.
24 and 81 can both be divided by 3.
So it is 3(8x³ + 27).
2 Write as cubes
Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.
8x³ is 2x cubed.
27 is 3 cubed.
So it is a sum of two cubes: a is 2x and b is 3.
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 = 2x and b = 3 into the pattern.
Careful!
It is not (2x + 3)³.
Multiplied out, that has x² and x terms too.
4 Tidy up
2x squared is 4x².
2x times 3 is 6x.
3 squared is 9.
So we get 3(2x + 3)(4x² − 6x + 9).
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 24x³ + 81 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.