A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 1: Algebraic methods · Lesson 3

Partial fractions

114 marks

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Date

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Quick questions to start.

A1
Needed today
Factorise fully $x^{3} - x^{2} - 6 x$ .
[2 marks]
Answer
A2
Last lesson
Write $\frac{3}{x - 1} + \frac{2}{x + 2}$ as a single fraction.
[2 marks]
Answer
A3
Needed today
$\frac{5 x + 1}{( x + 1 ) ^{2} ( x - 2 )}$ is to be split into partial fractions. Write down the form, using constants $A$ , $B$ and $C$ .
[2 marks]
Answer
A4
Needed today
Solve $A + B = 3$ and $2 A - B = 0$ .
[2 marks]
Answer
A5
Needed today
$7 x - 1 \equiv A (x - 1) + B (x + 2)$ . By choosing values of $x$ , find $A$ and $B$ .
[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 $\frac{5 x + 1}{( x - 1 ) ( x + 2 )}$ in partial fractions.

Hint Write

\frac{A}{x - 1} + \frac{B}{x + 2}

, multiply by

(x - 1) (x + 2)

, then put

x = 1

and

x = - 2

B1
Express $\frac{x + 11}{( x + 3 ) ( x - 1 )}$ in partial fractions.
[4 marks]
Answer

Example 2

Express $\frac{x ^{2} + 7 x - 6}{x ^{3} - x ^{2} - 2 x}$ in partial fractions.

Hint Factorise the bottom fully first:

x (x - 2) (x + 1)

. Three factors give three fractions.

B2
Express $\frac{2 x ^{2} + 9 x - 9}{x ^{3} - 9 x}$ in partial fractions.
[5 marks]
Answer

Example 3

Express $\frac{3 x ^{2} + x + 2}{( x + 1 ) ^{2} ( x - 3 )}$ in partial fractions.

Hint A repeated factor needs two fractions:

\frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}}

. Substitution gives two constants; compare the

x^{2}

terms for the last one.

B3
Express $\frac{x ^{2} + 5 x - 5}{( x - 2 ) ^{2} ( x + 1 )}$ in partial fractions.
[5 marks]
Answer

Example 4

Express $\frac{x + 5}{x ^{2} ( x + 1 )}$ in partial fractions.

Hint

x^{2}

is a repeated factor: use

\frac{A}{x} + \frac{B}{x ^{2}}

. Put

x = 0

and

x = - 1

, then compare the

x^{2}

terms.

B4
Express $\frac{3 x ^{2} - 2 x + 8}{x ^{2} ( x - 2 )}$ in partial fractions.
[5 marks]
Answer

Example 5

Express $\frac{8 x - 1}{( 2 x - 1 ) ( x + 1 )}$ in partial fractions.

Hint To make

2 x - 1

zero, put

x = \frac{1}{2}

B5
Express $\frac{7}{( 3 x + 1 ) ( x - 2 )}$ in partial fractions.
[4 marks]
Answer

Practice

A-Level

C1
Jo writes $\frac{5 x + 1}{( x - 1 ) ^{2} ( x + 2 )} \equiv \frac{A}{( x - 1 ) ^{2}} + \frac{B}{x + 2}$ . Show that no constants $A$ and $B$ make this true. Then express the fraction correctly in partial fractions.
[5 marks]
Answer

C2
Express $\frac{6 x ^{2} + 12 x}{( 1 - x ) ( 1 + 2 x ) ^{2}}$ in partial fractions.
[5 marks]
Answer
C3
Given that $\frac{2 x ^{2} + 2 x + 4}{( x - 1 ) ^{2} ( x + 3 )} \equiv \frac{A}{x - 1} + \frac{B}{( x - 1 ) ^{2}} + \frac{C}{x + 3}$ , find $A$ , $B$ and $C$ .
[5 marks]
Answer $A = \dots$ , $B = \dots$ , $C = \dots$
C4
$f (x) = \frac{4 x + 10}{( x + 1 ) ( x + 4 )}$ , $x > - 1$ . Express $f (x)$ in partial fractions. Hence show that $f^{'} (x) < 0$ for all $x > - 1$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Express $\frac{7 x ^{2} + 10 x}{2 x ^{3} + 3 x ^{2} - 1}$ in partial fractions.
[8 marks]
Answer

D2
$f (x) = \frac{6 + 5 x}{( 1 + 2 x ) ( 3 - x )}$ .
(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 the values of $x$ for which your expansion is valid.
[2 marks]
Answer
Total for D2: 10 marks
D3
Ben uses cover-up for every term of $\frac{9 x ^{2} + 5 x + 2}{x ^{2} ( x + 1 )}$ . For $\frac{A}{x}$ he covers the $x$ of $x^{2}$ and puts $x = 0$ , getting $A = 2$ . He writes $\frac{2}{x} + \frac{2}{x ^{2}} + \frac{6}{x + 1}$ .
(a)
By substituting $x = 1$ , show that Ben is wrong.
[3 marks]
(b)
Express $\frac{9 x ^{2} + 5 x + 2}{x ^{2} ( x + 1 )}$ correctly in partial fractions.
[5 marks]
Answer
Total for D3: 8 marks
D4
$g (x) = x^{3} + 2 x^{2} - 5 x - 6$ .
(a)
Use the factor theorem to factorise $g (x)$ fully.
[3 marks]
Answer
(b)
Express $\frac{3 x ^{2} + 5 x + 8}{g ( x )}$ in partial fractions.
[5 marks]
Answer
(c)
Hence show that $\frac{3 x ^{2} + 5 x + 8}{g ( x )} > 0$ for all $x > 2$ .
[3 marks]
Total for D4: 11 marks
D5
$\frac{p x + 18}{( x - 2 ) ( x + 4 )} \equiv \frac{4}{x - 2} + \frac{q}{x + 4}$ , where $p$ and $q$ are constants. Call the left side $f (x)$ .
(a)
Find $p$ and $q$ .
[4 marks]
Answer $p = \dots$ , $q = \dots$
(b)
Solve $f (x) = \frac{1}{x + 4}$ .
[3 marks]
Answer
(c)
Solve $f (x) = 0$ .
[2 marks]
Answer
Total for D5: 9 marks
D6
Given that $\frac{2 x ^{2} + k x - 4}{x ( x - 1 ) ( x + 2 )} \equiv \frac{2}{x} + \frac{A}{x - 1} + \frac{B}{x + 2}$ and that $A = 3$ , find $B$ and $k$ .
[6 marks]
Answer $B = \dots$ , $k = \dots$

Extension

Stretch

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

E1
Show that $\int_{1}^{3} \frac{3 x ^{2} + 5 x + 3}{x ^{2} ( x + 1 )} d x = 2 + ln 18$ .
[9 marks]
Hint 1 · What to try
$x^{2}$ is a repeated factor, so use $\frac{A}{x} + \frac{B}{x ^{2}} + \frac{C}{x + 1}$ .
Hint 2 · The first line
You get $\frac{2}{x} + \frac{3}{x ^{2}} + \frac{1}{x + 1}$ . The middle term integrates to $- \frac{3}{x}$ , not to a logarithm.
Hint 3 · The full method
Put in the limits, then use $ln a + ln b = ln ab$ and $2 ln 3 = ln 9$ to collect the logarithms.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can split a fraction with distinct linear factors into partial fractions.
I can choose the right form when a factor is repeated.
I can find the constants by substitution and by comparing coefficients.
I can use partial fractions to differentiate, expand as a series or integrate.

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