Step by step
The question
Question: expand (3x − 1)³.
Careful!
A power of a bracket is not the power of each part.
(3x − 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 3x 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, 3x to the power 3, −1 to the power 0.
That is 27x³.
Second term: 3 to the power 2 is 9.
3 times 9 times −1 is −27.
So −27x².
Third term: (−1) to the power 2 is 1.
3 times 3 is 9.
So 9x.
Fourth term: (−1) to the power 3 is −1.
So −1.
4 Add them up
So (3x − 1)³ is 27x³ − 27x² + 9x − 1.
5 Check
Let's check with x = 1.
The question: 2 to the power 3 is 8.
Our answer with x = 1: the terms add up to 8 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.