A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 8: The binomial expansion · Lesson 2

The binomial expansion

72 marks

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Date

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Quick questions to start.

A1
Last lesson
Work out $(3 8)$ without a calculator.
[2 marks]
Answer
A2
Last lesson
Find the coefficient of $x^{2}$ in $(1 + 3 x)^{4}$ .
[2 marks]
Answer
A3
From chapter 1
Simplify $(\frac{3}{x})^{2} \times (2 x)^{3}$ .
[2 marks]
Answer
A4
From chapter 2
Solve $4 k^{2} = 9 k$ , given that $k \neq = 0$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the first four terms, in ascending powers of $x$ , of $(1 - 3 x)^{9}$ .

Hint

(1 + b)^{n} = 1 + nb + (2 n) b^{2} + (3 n) b^{3} + \dots

with

b = - 3 x

B1
Find the first three terms, in ascending powers of $x$ , of $(2 + \frac{1}{4} x)^{10}$ .
[3 marks]
Answer

Example 2

Find the coefficient of $x^{5}$ in the expansion of $(2 - x)^{11}$ .

Hint The general term is

(r n) a^{n - r} b^{r}

. Here

r = 5

a = 2

b = - x

B2
Find the coefficient of $x^{6}$ in the expansion of $(1 - 2 x)^{10}$ .
[2 marks]
Answer

Example 3

Find the term independent of $x$ in the expansion of $(x + \frac{2}{x})^{6}$ .

Hint The general term holds

x^{6 - r} (\frac{2}{x})^{r}

, which is

x^{6 - 2 r}

. Make the power

0

B3
Find the term independent of $x$ in the expansion of $(2 x^{2} - \frac{1}{x})^{9}$ .
[3 marks]
Answer

Example 4

In the expansion of $(1 + k x)^{n}$ , the coefficient of $x$ is $15$ and the coefficient of $x^{2}$ is $90$ . Find $n$ and $k$ .

Hint

nk = 15

and

\frac{n ( n - 1 )}{2} k^{2} = 90

. Put

k = \frac{15}{n}

into the second.

B4
In the expansion of $(1 + k x)^{n}$ , the coefficient of $x$ is $- 24$ and the coefficient of $x^{2}$ is $252$ . Find $n$ and $k$ .
[4 marks]
Answer

Practice

A-Level

C1
Find the binomial expansion of $(2 x - \frac{1}{x})^{5}$ , giving each term in its simplest form.
[4 marks]
Answer

C2
In the expansion of $(1 + 2 x)^{n}$ , where $n \geq 3$ , the coefficient of $x^{3}$ is $4$ times the coefficient of $x^{2}$ . Find $n$ .
[4 marks]
Answer
C3
Find the coefficient of $x^{3}$ in the expansion of $(1 - x)^{2} (1 + 3 x)^{6}$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Ali writes: "The $x^{3}$ term of $(2 - \frac{x}{2})^{8}$ is $(3 8) 2^{5} (\frac{x}{2})$ , so the coefficient of $x^{3}$ is $56 \times 32 \times \frac{1}{2} = 896$ ."
Ali's answer is wrong.
(a)
Describe the two mistakes in Ali's working.
[2 marks]
(b)
Find the correct coefficient of $x^{3}$ .
[2 marks]
Answer
(c)
Hence find the coefficient of $x^{3}$ in the expansion of $(1 + 4 x) (2 - \frac{x}{2})^{8}$ .
[3 marks]
Answer
Total for D1: 7 marks

D2
$p$ is a positive constant.
(a)
Find the first three terms, in ascending powers of $x$ , of the binomial expansion of $(1 + p x)^{12}$ .
[2 marks]
Answer
(b)
In this expansion, the coefficient of $x^{2}$ is $22$ times the coefficient of $x$ . Find $p$ .
[3 marks]
Answer
(c)
Hence find the coefficient of $x^{3}$ in the expansion of $(3 - x) (1 + p x)^{12}$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
$a$ is a constant. In the expansion of $(a x - \frac{3}{x ^{2}})^{6}$ , the term independent of $x$ is $2160$ .
(a)
Find the two possible values of $a$ .
[4 marks]
Answer
(b)
Hence, for $a > 0$ , find the coefficient of $x^{3}$ in this expansion.
[3 marks]
Answer
(c)
Explain why there is no term in $x^{2}$ in this expansion.
[2 marks]
Total for D3: 9 marks
D4
In this question, do not use a calculator.
(a)
Show that $(x + y)^{5} - (x - y)^{5} = 10 x^{4} y + 20 x^{2} y^{3} + 2 y^{5}$ .
[3 marks]
(b)
Hence show that $(2 + 3)^{5} - (2 - 3)^{5} = k 3$ , and find the integer $k$ .
[3 marks]
Answer
(c)
Find the exact value of $(2 + 3)^{5} + (2 - 3)^{5}$ .
[4 marks]
Answer
Total for D4: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Find the coefficient of $x^{3}$ in the expansion of $(1 + x + x^{2})^{10}$ .
[5 marks]
Hint 1 · What to try
Write $1 + x + x^{2}$ as $1 + y$ , with $y = x + x^{2}$ .
Hint 2 · The first line
Only the $y^{2}$ and $y^{3}$ terms of $(1 + y)^{10}$ give $x^{3}$ .
Hint 3 · The full method
$y^{2}$ holds $2 x^{3}$ and $y^{3}$ holds $x^{3}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the first terms of a binomial expansion in ascending powers of x.
I can find one coefficient from the general term.
I can find a term independent of x when both terms hold x.
I can find an unknown constant or n from given coefficients.

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