A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 4: Binomial expansion · Lesson 3

Binomial expansion with partial fractions

123 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-binomial-expansion-with-partial-fractions/

Date

Do Now

Quick questions to start.

A1
Needed today
Express $\frac{x + 7}{( x + 1 ) ( x - 2 )}$ in partial fractions.
[3 marks]
Answer
A2
Needed today
Write down the first three terms of the expansions of $(1 - x)^{- 1}$ and of $(1 + 2 x)^{- 1}$ .
[2 marks]
Answer
A3
Common slip
Write $\frac{1}{x - 3}$ in the form $k (1 - \frac{x}{3})^{- 1}$ .
[2 marks]
Answer
A4
Common slip
One expansion is valid for $∣ x ∣ < \frac{1}{2}$ and another for $∣ x ∣ < 3$ . For which values of $x$ is their sum valid?
[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

Express $\frac{4 - 5 x}{( 1 + x ) ( 2 - x )}$ in partial fractions. Hence find its expansion in ascending powers of $x$ up to the term in $x^{3}$ , and state the values of $x$ for which it is valid.

Hint Expand

\frac{3}{1 + x}

and

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

separately, then subtract.

B1
Express $\frac{x + 5}{( 1 - x ) ( 2 + x )}$ in partial fractions. Hence find its expansion in ascending powers of $x$ up to the term in $x^{3}$ , and state the values of $x$ for which it is valid.
[7 marks]
Answer

Example 2

Given $\frac{1 + 3 x}{( 1 - x ) ( 1 + x ) ^{2}}$ $= \frac{1}{1 - x} + \frac{1}{1 + x} - \frac{1}{( 1 + x ) ^{2}}$ , find its expansion up to the term in $x^{3}$ .

Hint

(1 + x)^{- 2} = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots

B2
Express $\frac{3 - 4 x + 2 x ^{2}}{( 1 + x ) ( 1 - 2 x ) ^{2}}$ in the form $\frac{A}{1 + x} + \frac{B}{1 - 2 x} + \frac{C}{( 1 - 2 x ) ^{2}}$ . Hence find its expansion up to the term in $x^{2}$ and state the values of $x$ for which it is valid.
[8 marks]
Answer

Example 3

Express $\frac{3 x ^{2} + 4 x - 1}{( 1 - x ) ( 2 + x )}$ in the form $A + \frac{B}{1 - x} + \frac{C}{2 + x}$ . Hence find its expansion up to the term in $x^{2}$ .

Hint The top and bottom are both of degree 2, so there is a constant

A

. The

x^{2}

terms give

A

3 = - A

B3
Express $\frac{2 x ^{2} - x}{( 1 + x ) ( 2 - x )}$ in the form $A + \frac{B}{1 + x} + \frac{C}{2 - x}$ . Hence find its expansion up to the term in $x^{3}$ .
[8 marks]
Answer

Example 4

Using $\frac{4 - 5 x}{( 1 + x ) ( 2 - x )} = \frac{3}{1 + x} - \frac{2}{2 - x}$ , find the coefficient of $x^{r}$ in its expansion.

Hint The coefficient of

x^{r}

(1 + x)^{- 1}

(- 1)^{r}

, and in

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

it is

(\frac{1}{2})^{r}

B4
Using $\frac{x + 5}{( 1 - x ) ( 2 + x )} = \frac{2}{1 - x} + \frac{1}{2 + x}$ , find the coefficient of $x^{r}$ in its expansion.
[4 marks]
Answer

Practice

A-Level

C1
Mia uses $\frac{x + 4}{( x - 2 ) ( x + 1 )}$ $= \frac{2}{x - 2} - \frac{1}{x + 1}$ . She writes " $\frac{2}{x - 2} = 2 (1 + \frac{x}{2})^{- 1}$ $= 2 - x + \frac{1}{2} x^{2}$ ". Find her mistake and give the correct expansion of $\frac{x + 4}{( x - 2 ) ( x + 1 )}$ up to the term in $x^{2}$ .
[6 marks]
Answer

C2
$\frac{7}{( 1 - 2 x ) ( 3 + x )} = \frac{2}{1 - 2 x} + \frac{1}{3 + x}$ . Sam says the expansion of this is valid for $∣ x ∣ < 3$ . Is he right? Give the correct range and say whether the expansion could be used with $x = 0.4$ and with $x = 0.6$ .
[5 marks]
C3
$f (x) = \frac{3 + 3 x}{( 1 + 2 x ) ( 1 - x )}$ . (a) Express $f (x)$ in partial fractions and find its expansion up to the term in $x^{3}$ . (b) Use it to estimate $f (0.05)$ and find the percentage error, to 2 significant figures. (c) Explain why the expansion should not be used for $f (0.6)$ .
[8 marks]
C4
By writing $\frac{2 + x}{( 1 - x ) ^{2}} = \frac{A}{1 - x} + \frac{B}{( 1 - x ) ^{2}}$ , find the coefficient of $x^{r}$ in its expansion. Hence find the coefficient of $x^{5}$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
$f (x) = \frac{6 - 11 x}{( 2 + x ) ( 1 - 3 x )}$
(a)
Express $f (x)$ in partial fractions.
[3 marks]
Answer
(b)
Hence find the series expansion of $f (x)$ , in ascending powers of $x$ , up to and including the term in $x^{2}$ . Give each coefficient as a simplified fraction.
[5 marks]
Answer
(c)
State, with a reason, the values of $x$ for which the expansion is valid.
[2 marks]
Total for D1: 10 marks

D2
Lily writes: " $\frac{3}{1 - x} - \frac{2}{4 + x} = 3 (1 - x)^{- 1} - \frac{2}{4} (1 + x)^{- 1}$ , and this is valid for $∣ x ∣ < 4$ ."
Lily has made two errors.
(a)
Describe both errors.
[3 marks]
(b)
Find the correct expansion up to and including the term in $x^{2}$ .
[5 marks]
Answer
Total for D2: 8 marks
D3
Find the series expansion of $\frac{4 x ^{2} - 3 x + 2}{( 1 + x ) ( 1 - 2 x )}$ , in ascending powers of $x$ , up to and including the term in $x^{3}$ . State the values of $x$ for which it is valid.
[9 marks]
Answer
D4
$g (x) = \frac{5}{( 1 - 2 x ) ( 1 + 3 x )}$
(a)
Express $g (x)$ in partial fractions.
[2 marks]
Answer
(b)
Show that the coefficient of $x^{r}$ in the expansion of $g (x)$ is $2^{r + 1} + (- 1)^{r} 3^{r + 1}$ .
[4 marks]
(c)
Find the coefficient of $x^{4}$ and state the values of $x$ for which the expansion is valid.
[3 marks]
Answer
Total for D4: 9 marks
D5
$h (x) = \frac{10 + x}{( 1 + x ) ( 2 - x ) ^{2}}$
(a)
Express $h (x)$ in the form $\frac{A}{1 + x} + \frac{B}{2 - x} + \frac{C}{( 2 - x ) ^{2}}$ .
[4 marks]
Answer
(b)
Hence find the expansion of $h (x)$ up to and including the term in $x^{2}$ .
[4 marks]
Answer
(c)
Use your expansion to estimate $h (0.1)$ and find the percentage error, to 2 significant figures.
[2 marks]
Answer
Total for D5: 10 marks
D6
$p (x) = \frac{1 + k x}{( 1 - x ) ( 1 + 2 x )}$ , where $k$ is a constant.
(a)
Show that $p (x) = \frac{1 + k}{3 ( 1 - x )} + \frac{2 - k}{3 ( 1 + 2 x )}$ .
[3 marks]
(b)
In the expansion of $p (x)$ there is no term in $x^{2}$ . Find $k$ and the coefficient of $x^{3}$ .
[5 marks]
Answer
Total for D6: 8 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 expansion of $\frac{1}{( 1 - 2 x ) ( 1 + x )}$ is $\frac{2 ^{r + 1} + ( - 1 ) ^{r}}{3}$ . Prove that this is a whole number for every $r \geq 0$ . Find the smallest $r$ for which it is greater than $1000$ .
[10 marks]
Hint 1 · What to try
Start with partial fractions: $\frac{1}{( 1 - 2 x ) ( 1 + x )} = \frac{A}{1 - 2 x} + \frac{B}{1 + x}$ .
Hint 2 · The first line
For the whole number: $2 = 3 - 1$ , so $2^{r + 1}$ leaves the same remainder on division by $3$ as $(- 1)^{r + 1}$ .
Hint 3 · The full method
The coefficient roughly doubles each time. Try $r = 10$ and $r = 11$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can split a fraction into partial fractions and expand each part.
I can expand fractions with a repeated factor and improper fractions.
I can state the values of x for which the whole expansion is valid.
I can find the general term and use the expansion to approximate.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers