Factor 10x^3 + 10 Step by Step | Sum and difference of cubes

Step by step

The question

Question: factor 10x³ + 10.

10x³ + 10

1 Common factor first

Always look for a common factor first.

10 and 10 can both be divided by 10.

10, 10: all in the 10 times table

So it is 10(x³ + 1).

= 10(x³ + 1)

2 Write as cubes

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

x³ is x cubed.

1 is 1 cubed: 1 × 1 × 1 = 1.

cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

x³ = (x)³, 1 = 1³

So it is a sum of two cubes: a is x and b is 1.

= 10((x)³ + 1³)

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.

a³ + b³ = (a + b)(a² − ab + b²)

first bracket: same sign (+); middle: opposite (−); last: +

We put a = x and b = 1 into the pattern.

= 10[(x + 1)(x² − x × 1 + 1²)]

Careful!

It is not (x + 1)³.

Multiplied out, that has x² and x terms too.

Careful: not (x + 1)³ ✗

4 Tidy up

x squared is x².

x times 1 is x.

1 squared is 1.

(x)² = x², x × 1 = x, 1² = 1

So we get 10(x + 1)(x² − x + 1).

= 10(x + 1)(x² − x + 1)

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.

Check: (x + 1)(x² − x + 1)

= x³ − x² + x + x² − x + 1 = x³ + 1

Then the common factor times every part.

10(x³ + 1) = 10x³ + 10

We get 10x³ + 10 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.

Answer