Step by step
The question
Question: factor completely 3x³ − 27x.
Completely means: until no part can be factored any more.
1 Common factor first
All the numbers can be divided by 3.
Every term has x.
So the common factor is 3x.
We divide every term by it.
What is left goes in the bracket.
So it is 3x(x² − 9).
2 Count the terms
The bracket has 2 terms.
With 2 terms we look for two squares or two cubes.
x² − 9 is one square take away another square.
x² is x squared.
9 is 3 squared.
A square minus a square is the plus bracket times the minus bracket.
So we get 3x(x + 3)(x − 3).
3 Check each bracket
Brackets like x + 3 have just x and a number.
They can't be factored.
No bracket can be factored any more.
So we are done.
Careful!
Don't forget the common factor 3x in the answer.
4 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 3x³ − 27x again.
So it is right!
Let's look back.
First: common factor first.
Then: count the terms.
Then: check each bracket.
Last: check.