Expand (x + 1)^4 Step by Step | The binomial theorem

Step by step

The question

Question: expand (x + 1)⁴.

(x + 1)⁴

Careful!

A power of a bracket is not the power of each part.

(x + 1) to the power 4 is the bracket times itself 4 times.

There are middle terms too.

Careful: not (x + 1)⁴ = x⁴ + 1⁴ ✗

1 Use Pascal's triangle

Pascal's triangle starts with 1.

Every row starts and ends with 1.

Each other number is the two numbers above it added.

2 The pattern

There are 5 terms.

The numbers in front come from row 4: 1, 4, 6, 4, 1.

numbers in front : 1, 4, 6, 4, 1

The powers of x go down from 4 to 0.

powers of x : 4, 3, 2, 1, 0

The powers of 1 go up from 0 to 4.

powers of (1) : 0, 1, 2, 3, 4

Remember: any number to the power 0 is 1.

any number⁰ = 1

3 Work out each term

The first term: 1, x to the power 4, 1 to the power 0.

That is x⁴.

1 × x⁴ × 1⁰ = x⁴

Second term: So 4x³.

4 × x³ × 1¹ = 4x³

Third term: 1 to the power 2 is 1.

So 6x².

6 × x² × 1² = 6x²

Fourth term: 1 to the power 3 is 1.

So 4x.

4 × x¹ × 1³ = 4x

Fifth term: 1 to the power 4 is 1.

So 1.

1 × x⁰ × 1⁴ = 1

4 Add them up

So (x + 1)⁴ is x⁴ + 4x³ + 6x² + 4x + 1.

(x + 1)⁴ = x⁴ + 4x³ + 6x² + 4x + 1

5 Check

Let's check with x = 1.

The question: 2 to the power 4 is 16.

x = 1: (1 + 1)⁴ = 2⁴ = 16

Our answer with x = 1: the terms add up to 16 too.

It works!

answer: 1 + 4 + 6 + 4 + 1 = 16

Let's look back.

First: use Pascal's triangle.

Then: the pattern.

Then: work out each term.

Then: add them up.

Last: check.

Answer