Step by step
The question
Question: expand (x − 3)⁴.
Careful!
A power of a bracket is not the power of each part.
(x − 3) to the power 4 is the bracket times itself 4 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 5 terms.
The numbers in front come from row 4: 1, 4, 6, 4, 1.
The powers of x go down from 4 to 0.
The powers of −3 go up from 0 to 4.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, x to the power 4, −3 to the power 0.
That is x⁴.
Second term: 4 times −3 is −12.
So −12x³.
Third term: (−3) to the power 2 is 9.
6 times 9 is 54.
So 54x².
Fourth term: (−3) to the power 3 is −27.
4 times −27 is −108.
So −108x.
Fifth term: (−3) to the power 4 is 81.
So 81.
4 Add them up
So (x − 3)⁴ is x⁴ − 12x³ + 54x² − 108x + 81.
5 Check
Let's check with x = 1.
The question: (−2) to the power 4 is 16.
Our answer with x = 1: the terms add up to 16 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.