Step by step
The question
Question: factor x³ + 27.
1 Write as cubes
Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.
x³ is x cubed.
27 is 3 cubed.
So it is a sum of two cubes: a is x and b is 3.
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 = 3 into the pattern.
Careful!
It is not (x + 3)³.
Multiplied out, that has x² and x terms too.
3 Tidy up
x squared is x².
x times 3 is 3x.
3 squared is 9.
So we get (x + 3)(x² − 3x + 9).
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 + 27 again.
So it is right!
Let's look back.
First: write as cubes.
Then: use the pattern.
Then: tidy up.
Last: check.