Factor 2x^3 + 250

Step by step

The question

Question: factor 2x³ + 250.

1 Common factor first

Always look for a common factor first.

2 and 250 can both be divided by 2.

So it is 2(x³ + 125).

2 Write as cubes

Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.

x³ is x cubed.

125 is 5 cubed.

So it is a sum of two cubes: a is x and b is 5.

3 Use the pattern

Here is the pattern.

The first bracket has the same sign as the question.

In the second bracket the middle sign is the opposite one, and the last sign is always plus.

We put a = x and b = 5 into the pattern.

Careful!

It is not (x + 5)³.

Multiplied out, that has x² and x terms too.

4 Tidy up

x squared is x².

x times 5 is 5x.

5 squared is 25.

So we get 2(x + 5)(x² − 5x + 25).

The second bracket can't be factored: with this pattern it never can.

5 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 2x³ + 250 again.

So it is right!

Let's look back.

First: common factor first.

Then: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.