Step by step
The question
Question: expand (2x − 1)⁴.
Careful!
A power of a bracket is not the power of each part.
(2x − 1) 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 2x go down from 4 to 0.
The powers of −1 go up from 0 to 4.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, 2x to the power 4, −1 to the power 0.
That is 16x⁴.
Second term: 2 to the power 3 is 8.
4 times 8 times −1 is −32.
So −32x³.
Third term: 2 to the power 2 is 4.
(−1) to the power 2 is 1.
6 times 4 is 24.
So 24x².
Fourth term: (−1) to the power 3 is −1.
4 times 2 times −1 is −8.
So −8x.
Fifth term: (−1) to the power 4 is 1.
So 1.
4 Add them up
So (2x − 1)⁴ is 16x⁴ − 32x³ + 24x² − 8x + 1.
5 Check
Let's check with x = 1.
The question: 1 to the power 4 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.