In the expansion of (a + b)⁶, what do the powers of a and b add up to in each term?

6. As one power goes down, the other goes up.

The binomial expansion · Cambridge IGCSE Additional Mathematics

In plain words

Multiplying out (a + b)⁵ by hand means five brackets and a lot of collecting. The binomial theorem gives the answer directly: the powers follow a simple pattern, and the numbers in front are combinations.

6 things to know

(a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + bⁿ. This is on the formula list.
The power of a goes down by one each term, and the power of b goes up by one. The two powers always add up to n.
The general term is nCr × aⁿ⁻ʳ × bʳ. It is the term that contains bʳ.
There are n + 1 terms in the expansion.
Keep each of a and b in brackets while you work, especially when one of them is negative or has a number in front, as in (2x)³ = 8x³.
The term independent of x is the one in which the powers of x cancel out.

Worked example

Expand (2 + x)⁴.

The coefficients 4C0 to 4C4 are 1, 4, 6, 4, 1.
(2 + x)⁴ = 2⁴ + 4 × 2³ × x + 6 × 2² × x² + 4 × 2 × x³ + x⁴.
= 16 + 32x + 24x² + 8x³ + x⁴.

Worked example

Find the term independent of x in the expansion of (x + 2/x)⁶.

The general term is 6Cr × x⁶⁻ʳ × (2/x)ʳ = 6Cr × 2ʳ × x⁶⁻²ʳ.
The power of x is 0 when 6 − 2r = 0, so r = 3.
6C3 × 2³ = 20 × 8 = 160.

Tips and tricks

Powers apply to the whole bracket: (2x)³ is 8x³, not 2x³. And (−x)² is positive.
To find one particular term, use the general term. There is no need to write out the whole expansion.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The binomial expansion: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

How many terms are there in the expansion of (a + b)⁷?
- 6
- 7
- 8 (the answer)
- 14
One more than the power.
What is the coefficient of x³ in the expansion of (1 + 2x)⁵?
- 10
- 40
- 80 (the answer)
- 160
5C3 × 2³ = 10 × 8.
What is (2x)³?
- 2x³
- 6x³
- 8x³ (the answer)
- 8x
The 2 is cubed as well.
In the expansion of (a + b)⁶, what do the powers of a and b add up to in each term?
- 5
- 6 (the answer)
- 7
- 12
As one power goes down, the other goes up.
What is the coefficient of x in the expansion of (3 + x)⁴?
- 12
- 27
- 108 (the answer)
- 81
4C1 × 3³ = 4 × 27.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 10 marks. Write your answers on paper, then check them.