Factor completely x^3 - 25x Step by Step

Step by step

The question

Question: factor completely x³ − 25x.

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

x³ − 25x

1 Common factor first

Every term has x.

So the common factor is x.

every term has x

We divide every term by it.

What is left goes in the bracket.

x³ ÷ x = x²

−25x ÷ x = −25

So it is x(x² − 25).

= x(x² − 25)

2 Count the terms

The bracket has 2 terms.

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

x² − 25 is one square take away another square.

x² is x squared.

25 is 5 squared.

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

a² − b² = (a + b)(a − b)

x² = (x)², 25 = 5²

So we get x(x + 5)(x − 5).

= x(x + 5)(x − 5)

3 Check each bracket

Brackets like x + 5 have just x and a number.

They can't be factored.

x + 5, x − 5: just x and a number

No bracket can be factored any more.

So we are done.

Careful!

Don't forget the common factor x in the answer.

Careful: not (x² − 25) ✗

4 Check

Let's check: we multiply it back out.

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

Check: (x + 5)(x − 5) = x² − 5x + 5x − 25

= x² − 25

Then the common factor times every part.

x(x² − 25) = x³ − 25x

We get x³ − 25x again.

So it is right!

Let's look back.

First: common factor first.

Then: count the terms.

Then: check each bracket.

Last: check.

Answer