Expand (x + 1)^6 Step by Step | The binomial theorem

Step by step

The question

Question: expand (x + 1)⁶.

(x + 1)⁶

Careful!

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

(x + 1) to the power 6 is the bracket times itself 6 times.

There are middle terms too.

Careful: not (x + 1)⁶ = x⁶ + 1⁶ ✗

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 7 terms.

The numbers in front come from row 6: 1, 6, 15, 20, 15, 6, 1.

numbers in front : 1, 6, 15, 20, 15, 6, 1

The powers of x go down from 6 to 0.

powers of x : 6, 5, 4, 3, 2, 1, 0

The powers of 1 go up from 0 to 6.

powers of (1) : 0, 1, 2, 3, 4, 5, 6

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 6, 1 to the power 0.

That is x⁶.

1 × x⁶ × 1⁰ = x⁶

Second term: So 6x⁵.

6 × x⁵ × 1¹ = 6x⁵

Third term: 1 to the power 2 is 1.

So 15x⁴.

15 × x⁴ × 1² = 15x⁴

Fourth term: 1 to the power 3 is 1.

So 20x³.

20 × x³ × 1³ = 20x³

Fifth term: 1 to the power 4 is 1.

So 15x².

15 × x² × 1⁴ = 15x²

Sixth term: 1 to the power 5 is 1.

So 6x.

6 × x¹ × 1⁵ = 6x

Seventh term: 1 to the power 6 is 1.

So 1.

1 × x⁰ × 1⁶ = 1

4 Add them up

So (x + 1)⁶ is x⁶ + 6x⁵ + 15x⁴ + 20x³ + 15x² + 6x + 1.

(x + 1)⁶ = x⁶ + 6x⁵ + 15x⁴ + 20x³ + 15x² + 6x + 1

5 Check

Let's check with x = 1.

The question: 2 to the power 6 is 64.

x = 1: (1 + 1)⁶ = 2⁶ = 64

Our answer with x = 1: the terms add up to 64 too.

It works!

answer: 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64

Let's look back.

First: use Pascal's triangle.

Then: the pattern.

Then: work out each term.

Then: add them up.

Last: check.

Answer