Factor x^2 + x + 1

Step by step

The question

Question: factor completely x² + x + 1.

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 3 terms.

With 3 terms we look for two numbers that multiply to c and add to b.

3 Prime or not

We need two numbers.

They multiply to 1, and they add to 1.

We list every pair that makes 1.

We add each pair.

We need 1.

None of them gives 1.

No pair works.

So x² + x + 1 does not factor with whole numbers.

We call it prime.

So x² + x + 1 is prime.

Let's look back.

First: common factor first.

Then: count the terms.

Last: prime or not.