Factor completely 3x^4 - 3 Step by Step

Step by step

The question

Question: factor completely 3x⁴ − 3.

Completely means: until no part can be factored any more.

3x⁴ − 3

1 Common factor first

All the numbers can be divided by 3.

So the common factor is 3.

3, 3: all in the 3 times table

We divide every term by it.

What is left goes in the bracket.

3x⁴ ÷ 3 = x⁴

−3 ÷ 3 = −1

So it is 3(x⁴ − 1).

= 3(x⁴ − 1)

2 Count the terms

The bracket has 2 terms.

With 2 terms we look for two squares or two cubes.

x⁴ − 1 is one square take away another square.

x⁴ is x² squared.

1 is 1 squared.

A square minus a square is the plus bracket times the minus bracket.

a² − b² = (a + b)(a − b)

x⁴ = (x²)², 1 = 1²

So we get 3(x² + 1)(x² − 1).

= 3(x² + 1)(x² − 1)

3 Factor again

x² − 1 is one square take away another square.

x² is x squared.

1 is 1 squared.

A square minus a square is the plus bracket times the minus bracket.

a² − b² = (a + b)(a − b)

x² = (x)², 1 = 1²

So we get 3(x² + 1)(x + 1)(x − 1).

= 3(x² + 1)(x + 1)(x − 1)

4 Check each bracket

Brackets like x + 1 have just x and a number.

They can't be factored.

x + 1, x − 1: just x and a number

x² + 1 is a square PLUS a square.

A sum of two squares does not factor.

We call it prime.

a² + b² → prime

No bracket can be factored any more.

So we are done.

Careful!

That is not the end: x² − 1 is two squares again.

Careful: not (x² + 1)(x² − 1) ✗

5 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (x + 1)(x − 1) = x² − x + x − 1

= x² − 1

Then the next bracket.

(x² + 1)(x² − 1) = x⁴ − x² + x² − 1 = x⁴ − 1

Then the common factor times every part.

3(x⁴ − 1) = 3x⁴ − 3

We get 3x⁴ − 3 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.

Answer