Step by step
The question
Question: factor completely 6x³ − 24x² + 24x.
Completely means: until no part can be factored any more.
1 Common factor first
All the numbers can be divided by 6.
Every term has x.
So the common factor is 6x.
We divide every term by it.
What is left goes in the bracket.
So it is 6x(x² − 4x + 4).
2 Count the terms
The bracket has 3 terms.
With 3 terms we look for two numbers that multiply to c and add to b.
We need two numbers.
They multiply to 4, and they add to −4.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 4, smallest first.
We add each pair.
We need 4.
2 and 2 give 4.
Yes!
With the signs: −2 and −2.
So we get 6x(x − 2)².
3 Check each bracket
Brackets like x − 2 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 6x 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 6x³ − 24x² + 24x again.
So it is right!
Let's look back.
First: common factor first.
Then: count the terms.
Then: check each bracket.
Last: check.