Worked examples
The question
Question: expand (x + 1)³.
(x + 1)3
Careful!
A power of a bracket is not the power of each part.
(x + 1) to the power 3 is the bracket times itself 3 times.
There are middle terms too.
Careful: not (x + 1)3 = x3 + 13 ✗
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.
numbers in front : 1, 3, 3, 1
The powers of x go down from 3 to 0.
powers of x : 3, 2, 1, 0
The powers of 1 go up from 0 to 3.
powers of (1) : 0, 1, 2, 3
Remember: any number to the power 0 is 1.
any number0 = 1
3 Work out each term
The first term: 1, x to the power 3, 1 to the power 0.
That is x³.
1 × x3 × 10 = x3
Second term: So 3x².
3 × x2 × 11 = 3x2
Third term: 1 to the power 2 is 1.
So 3x.
3 × x1 × 12 = 3x
Fourth term: 1 to the power 3 is 1.
So 1.
1 × x0 × 13 = 1
4 Add them up
So (x + 1)³ is x³ + 3x² + 3x + 1.
(x + 1)3 = x3 + 3x2 + 3x + 1
5 Check
Let's check with x = 1.
The question: 2 to the power 3 is 8.
x = 1: (1 + 1)3 = 23 = 8
Our answer with x = 1: the terms add up to 8 too.
It works!
answer: 1 + 3 + 3 + 1 = 8
Let's look back.
First: use Pascal's triangle.
Then: the pattern.
Then: work out each term.
Then: add them up.
Last: check.
Answer (x + 1)3 = x3 + 3x2 + 3x + 1
▶ Watch it solved
The question
Question: expand (x − 1)⁵.
(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.
Careful: not (x − 1)5 = x5 − 15 ✗
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.
numbers in front : 1, 5, 10, 10, 5, 1
The powers of x go down from 5 to 0.
powers of x : 5, 4, 3, 2, 1, 0
The powers of −1 go up from 0 to 5.
powers of (−1) : 0, 1, 2, 3, 4, 5
Remember: any number to the power 0 is 1.
any number0 = 1
3 Work out each term
The first term: 1, x to the power 5, −1 to the power 0.
That is x⁵.
1 × x5 × (−1)0 = x5
Second term: 5 times −1 is −5.
So −5x⁴.
5 × x4 × (−1)1 = 5 × (−1) × x4 = −5x4
Third term: (−1) to the power 2 is 1.
So 10x³.
10 × x3 × (−1)2 = 10x3
Fourth term: (−1) to the power 3 is −1.
10 times −1 is −10.
So −10x².
10 × x2 × (−1)3 = 10 × (−1) × x2 = −10x2
Fifth term: (−1) to the power 4 is 1.
So 5x.
5 × x1 × (−1)4 = 5x
Sixth term: (−1) to the power 5 is −1.
So −1.
1 × x0 × (−1)5 = −1
4 Add them up
So (x − 1)⁵ is x⁵ − 5x⁴ + 10x³ − 10x² + 5x − 1.
(x − 1)5 = x5 − 5x4 + 10x3 − 10x2 + 5x − 1
5 Check
Let's check with x = 2.
The question: 1 to the power 5 is 1.
x = 2: (2 − 1)5 = 15 = 1
Our answer with x = 2: the terms add up to 1 too.
It works!
answer: 32 − 80 + 80 − 40 + 10 − 1 = 1
Let's look back.
First: use Pascal's triangle.
Then: the pattern.
Then: work out each term.
Then: add them up.
Last: check.
Answer (x − 1)5 = x5 − 5x4 + 10x3 − 10x2 + 5x − 1
▶ Watch it solved
The question
Question: expand (2x − 1)⁴.
(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.
Careful: not (2x − 1)4 = 2x4 + 14 ✗
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.
numbers in front : 1, 4, 6, 4, 1
The powers of 2x go down from 4 to 0.
powers of 2x : 4, 3, 2, 1, 0
The powers of −1 go up from 0 to 4.
powers of (−1) : 0, 1, 2, 3, 4
Remember: any number to the power 0 is 1.
any number0 = 1
3 Work out each term
The first term: 1, 2x to the power 4, −1 to the power 0.
That is 16x⁴.
1 × (2x)4 × (−1)0 = 16x4
Second term: 2 to the power 3 is 8.
4 times 8 times −1 is −32.
So −32x³.
4 × (2x)3 × (−1)1 = 4 × 8 × (−1) × x3 = −32x3
Third term: 2 to the power 2 is 4.
(−1) to the power 2 is 1.
6 times 4 is 24.
So 24x².
6 × (2x)2 × (−1)2 = 6 × 4 × x2 = 24x2
Fourth term: (−1) to the power 3 is −1.
4 times 2 times −1 is −8.
So −8x.
4 × (2x)1 × (−1)3 = 4 × 2 × (−1) × x = −8x
Fifth term: (−1) to the power 4 is 1.
So 1.
1 × (2x)0 × (−1)4 = 1
4 Add them up
So (2x − 1)⁴ is 16x⁴ − 32x³ + 24x² − 8x + 1.
(2x − 1)4 = 16x4 − 32x3 + 24x2 − 8x + 1
5 Check
Let's check with x = 1.
The question: 1 to the power 4 is 1.
x = 1: (2 × 1 − 1)4 = 14 = 1
Our answer with x = 1: the terms add up to 1 too.
It works!
answer: 16 − 32 + 24 − 8 + 1 = 1
Let's look back.
First: use Pascal's triangle.
Then: the pattern.
Then: work out each term.
Then: add them up.
Last: check.
Answer (2x − 1)4 = 16x4 − 32x3 + 24x2 − 8x + 1
▶ Watch it solved
