Step by step
The question
Question: expand (x − 1)⁵.
Careful!
A power of a bracket is not the power of each part.
(x − 1) to the power 5 is the bracket times itself 5 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 6 terms.
The numbers in front come from row 5: 1, 5, 10, 10, 5, 1.
The powers of x go down from 5 to 0.
The powers of −1 go up from 0 to 5.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, x to the power 5, −1 to the power 0.
That is x⁵.
Second term: 5 times −1 is −5.
So −5x⁴.
Third term: (−1) to the power 2 is 1.
So 10x³.
Fourth term: (−1) to the power 3 is −1.
10 times −1 is −10.
So −10x².
Fifth term: (−1) to the power 4 is 1.
So 5x.
Sixth term: (−1) to the power 5 is −1.
So −1.
4 Add them up
So (x − 1)⁵ is x⁵ − 5x⁴ + 10x³ − 10x² + 5x − 1.
5 Check
Let's check with x = 2.
The question: 1 to the power 5 is 1.
Our answer with x = 2: the terms add up to 1 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.