Step by step
The question
Question: factor completely x⁴ − 81.
Completely means: until no part can be factored any more.
1 Common factor first
Always ask this first.
Here no number and no x goes into every term.
So there is no common factor.
2 Count the terms
The bracket has 2 terms.
With 2 terms we look for two squares or two cubes.
x⁴ − 81 is one square take away another square.
x⁴ is x² squared.
81 is 9 squared.
A square minus a square is the plus bracket times the minus bracket.
So we get (x² + 9)(x² − 9).
3 Factor again
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 (x² + 9)(x + 3)(x − 3).
4 Check each bracket
Brackets like x + 3 have just x and a number.
They can't be factored.
x² + 9 is a square PLUS a square.
A sum of two squares does not factor.
We call it prime.
No bracket can be factored any more.
So we are done.
Careful!
That is not the end: x² − 9 is two squares again.
5 Check
Let's check: we multiply it back out.
Each part of one bracket times each part of the other.
Then the next bracket.
We get x⁴ − 81 again.
So it is right!
Let's look back.
First: common factor first.
Then: count the terms.
Then: factor again.
Then: check each bracket.
Last: check.