Step by step
The question
Question: expand (2x + 1)³.
Careful!
A power of a bracket is not the power of each part.
(2x + 1) to the power 3 is the bracket times itself 3 times.
There are middle terms too.
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 4 terms.
The numbers in front come from row 3: 1, 3, 3, 1.
The powers of 2x go down from 3 to 0.
The powers of 1 go up from 0 to 3.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, 2x to the power 3, 1 to the power 0.
That is 8x³.
Second term: 2 to the power 2 is 4.
3 times 4 is 12.
So 12x².
Third term: 1 to the power 2 is 1.
3 times 2 is 6.
So 6x.
Fourth term: 1 to the power 3 is 1.
So 1.
4 Add them up
So (2x + 1)³ is 8x³ + 12x² + 6x + 1.
5 Check
Let's check with x = 1.
The question: 3 to the power 3 is 27.
Our answer with x = 1: the terms add up to 27 too.
It works!
Let's look back.
First: use Pascal's triangle.
Then: the pattern.
Then: work out each term.
Then: add them up.
Last: check.