Step by step
The question
Question: expand (x − 2)⁵.
Careful!
A power of a bracket is not the power of each part.
(x − 2) 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 −2 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, −2 to the power 0.
That is x⁵.
Second term: 5 times −2 is −10.
So −10x⁴.
Third term: (−2) to the power 2 is 4.
10 times 4 is 40.
So 40x³.
Fourth term: (−2) to the power 3 is −8.
10 times −8 is −80.
So −80x².
Fifth term: (−2) to the power 4 is 16.
5 times 16 is 80.
So 80x.
Sixth term: (−2) to the power 5 is −32.
So −32.
4 Add them up
So (x − 2)⁵ is x⁵ − 10x⁴ + 40x³ − 80x² + 80x − 32.
5 Check
Let's check with x = 1.
The question: (−1) to the power 5 is −1.
Our answer with x = 1: 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.