A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 4: Binomial expansion · Lesson 1

Expanding (1 + x)ⁿ for any n

117 marks

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Date

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Quick questions to start.

A1
Needed today
Expand $(1 + 2 x)^{4}$ fully.
[3 marks]
Answer
A2
Needed today
Work out $\frac{( - \frac{1}{2} ) ( - \frac{3}{2} ) ( - \frac{5}{2} )}{3 !}$ .
[2 marks]
Answer
A3
Common slip
Simplify $(- 3 x)^{2}$ and $(- 3 x)^{3}$ .
[2 marks]
Answer
A4
Needed today
Write $\frac{1}{1 + x}$ and $\frac{1}{( 1 - x ) ^{2}}$ as powers of a bracket.
[2 marks]
Answer
A5
Common slip
Solve $∣3 x ∣ < 1$ .
[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 of the binomial expansion of $(1 - x)^{- 3}$ , in ascending powers of $x$ . State the values of $x$ for which the expansion is valid.

Hint Use

(1 + y)^{n} = 1 + n y + \frac{n ( n - 1 )}{2 !} y^{2}

+ \frac{n ( n - 1 ) ( n - 2 )}{3 !} y^{3} + \dots

with

n = - 3

and

y = - x

B1
Find the first four terms of the binomial expansion of $(1 + x)^{- 4}$ , in ascending powers of $x$ . State the values of $x$ for which it is valid.
[4 marks]
Answer

Example 2

Find the first four terms of the expansion of $1 - 4 x$ and state the values of $x$ for which it is valid.

Hint Write it as

(1 - 4 x)^{\frac{1}{2}}

. Keep

(- 4 x)

in a bracket every time you raise it to a power.

B2
Find the first four terms of the expansion of $(1 + 3 x)^{- \frac{1}{3}}$ and state the values of $x$ for which it is valid.
[5 marks]
Answer

Example 3

Use your expansion of $1 - 4 x$ from the example above, with $x = 0.01$ , to find an approximation to $6$ , to 5 decimal places.

Hint Work out

1 - 4 (0.01) = 0.96

and write it in terms of

6

B3
Use your expansion of $(1 + 3 x)^{- \frac{1}{3}}$ from B2 with $x = - \frac{1}{27}$ to find an approximation to $39$ , to 3 decimal places.
[3 marks]
Answer

Example 4

The first three terms of the expansion of $(1 + k x)^{n}$ are $1 - 6 x + 27 x^{2}$ . Find $k$ and $n$ .

Hint Write

nk = - 6

and

\frac{n ( n - 1 )}{2} k^{2} = 27

. Use the first equation to remove

k

from the second.

B4
The first three terms of the expansion of $(1 + k x)^{n}$ are $1 - 4 x - 8 x^{2}$ . Find $k$ and $n$ , the term in $x^{3}$ , and the values of $x$ for which the expansion is valid.
[6 marks]
Answer

Example 5

Find the expansion of $(2 - x) 1 + 4 x$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .

Hint Expand

1 + 4 x

to the

x^{2}

term, then multiply by

(2 - x)

and collect only up to

x^{2}

B5
Find the expansion of $\frac{3 + x}{1 - 2 x}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .
[4 marks]
Answer

Practice

A-Level

C1
Mia expands $(1 - 2 x)^{- \frac{1}{2}}$ . She writes: " $1 + (- \frac{1}{2}) (- 2 x) + \frac{( - \frac{1}{2} ) ( - \frac{3}{2} )}{2 !} (- 2 x^{2}) = 1 + x - \frac{3}{4} x^{2}$ ." Find her mistake and give the correct first four terms.
[4 marks]
Answer

C2
Find the first three terms of the expansion of $(1 + 5 x)^{- 1}$ and state the values of $x$ for which it is valid. Ben substitutes $x = 0.5$ into the series to find $\frac{1}{3.5}$ . Explain why this does not work.
[4 marks]
Answer
C3
(a) Find the first three terms of the expansion of $(1 - 8 x)^{\frac{1}{2}}$ . (b) By substituting $x = 0.01$ , find an approximation to $23$ , to 3 decimal places. (c) Explain why $x = 0.25$ cannot be used in the expansion.
[6 marks]
Answer
C4
In the expansion of $(1 + a x)^{- 3}$ , where $a$ is a non-zero constant, the coefficient of $x^{3}$ is equal to the coefficient of $x^{2}$ . Find $a$ and state the values of $x$ for which the expansion is valid.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
$f (x) = (1 - 9 x)^{- \frac{1}{3}}$
(a)
Find the first four terms of the binomial expansion of $f (x)$ , in ascending powers of $x$ .
[4 marks]
Answer
(b)
Show that $f (- \frac{2}{225}) = \frac{3 25}{3}$ , and explain why $x = - \frac{2}{225}$ is a suitable value to use in the expansion.
[3 marks]
(c)
Hence find an approximation to $325$ , giving your answer to 4 decimal places.
[3 marks]
Answer
Total for D1: 10 marks

D2
Leo writes: " $(1 + 2 x)^{- 3} = 1 - 6 x + 24 x^{2} - 80 x^{3} + \dots$ , and this is true for all values of $x$ ."
Leo's terms are correct, but his statement is wrong.
(a)
Explain why Leo's statement is wrong, and state the correct values of $x$ .
[2 marks]
(b)
Given that $(a + b x) (1 + 2 x)^{- 3} = 2 + c x^{2} + \dots$ , where the expansion has no term in $x$ , find the values of the constants $a$ , $b$ and $c$ .
[6 marks]
Answer
Total for D2: 8 marks
D3
Find the binomial expansion of $\frac{1 + 2 x}{1 - 2 x}$ , in ascending powers of $x$ , up to and including the term in $x^{3}$ . By substituting $x = \frac{1}{20}$ , find an approximation to $11$ , giving your answer to 3 decimal places.
[9 marks]
Answer
D4
In the binomial expansion of $(1 + k x)^{n}$ , where $k$ and $n$ are constants and $n$ is not a positive integer, the coefficients of $x$ , $x^{2}$ and $x^{3}$ are in the ratio $2 : 9 : 45$ .
(a)
Find the value of $n$ and the value of $k$ .
[5 marks]
Answer
(b)
Find the coefficient of $x$ and the coefficient of $x^{3}$ .
[2 marks]
Answer
(c)
State the values of $x$ for which the expansion is valid, and explain why it cannot be used to find $(1 + k x)^{n}$ when $x = \frac{1}{5}$ .
[2 marks]
Total for D4: 9 marks
D5
$g (x) = (1 + 4 x)^{\frac{3}{2}}$
(a)
Find the first four terms of the binomial expansion of $g (x)$ , in ascending powers of $x$ .
[4 marks]
Answer
(b)
Use your expansion with $x = 0.05$ to find an estimate of $1. 2^{\frac{3}{2}}$ . Find the percentage error of this estimate, giving your answer to 2 significant figures.
[4 marks]
Answer
(c)
Explain why using $x = 0.3$ in the expansion would not give a reliable estimate of $2. 2^{\frac{3}{2}}$ .
[2 marks]
Total for D5: 10 marks
D6
$h (x) = \frac{2 + x}{1 + x}$ , $∣ x ∣ < 1$
(a)
Show that, for small values of $x$ , $h (x) \approx 2 + \frac{1}{4} x^{2}$ .
[4 marks]
(b)
Use this approximation to estimate $h (0.1)$ .
[2 marks]
Answer
(c)
Find the term in $x^{3}$ in the expansion of $h (x)$ .
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

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

E1
Show that the coefficient of $x^{r}$ in the binomial expansion of $(1 - 4 x)^{- \frac{1}{2}}$ is $(r 2 r)$ . Hence prove that, for every positive integer $r$ , $k = 0 \sum r (k 2 k) (r - k 2 r - 2 k) = 4^{r}$ .
[10 marks]
Hint 1 · What to try
Write the general coefficient as $\frac{n ( n - 1 ) \dots ( n - r + 1 )}{r !} (- 4)^{r}$ with $n = - \frac{1}{2}$ , and take out a factor of $(- \frac{1}{2})^{r}$ .
Hint 2 · The first line
The odd numbers $1 \times 3 \times 5 \times \dots \times (2 r - 1)$ equal $\frac{( 2 r )!}{2 ^{r} r !}$ , because $(2 r)!$ is the odd numbers times the even numbers $2 \times 4 \times \dots \times 2 r = 2^{r} r!$ .
Hint 3 · The full method
Square the whole series: $((1 - 4 x)^{- \frac{1}{2}})^{2} = (1 - 4 x)^{- 1}$ . Compare the coefficient of $x^{r}$ on each side.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the first four terms of (1 + bx)ⁿ when n is negative or a fraction.
I can state the values of x for which the expansion is valid and explain why.
I can choose a value of x to approximate a root and say how accurate it is.
I can find unknown constants from given coefficients.

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