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

Algebraic division and improper fractions

111 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Divide $x^{3} + 2 x^{2} - 5 x + 6$ by $(x + 4)$ . Give the quotient and the remainder.
[3 marks]
Answer
A2
Needed today
Is $\frac{x ^{2} + 1}{x ^{2} - 1}$ a proper or an improper fraction? Give a reason.
[2 marks]
Answer
A3
Last lesson
Express $\frac{5 x - 1}{( x + 1 ) ( x - 2 )}$ in partial fractions.
[3 marks]
Answer
A4
Needed today
Find $A$ and $B$ such that $\frac{2 x + 5}{x + 1} \equiv A + \frac{B}{x + 1} .$
[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

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

Hint The top and bottom have the same degree, so start with a whole number:

\frac{3 x ^{2} - 2 x - 2}{x ( x - 1 )} \equiv A + \frac{B}{x} + \frac{C}{x - 1}

B1
Express $\frac{2 x ^{2} - x + 5}{( x + 1 ) ( x - 3 )}$ in partial fractions.
[5 marks]
Answer

Example 2

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

Hint The top has degree

3

and the bottom degree

2

, so divide first. The quotient is linear.

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

Example 3

Express $\frac{4 x ^{2} + x}{( 2 x - 1 ) ( x + 1 )}$ in partial fractions.

Hint Multiply out the bottom:

2 x^{2} + x - 1

. The whole number part is

\frac{4}{2} = 2

, not

4

B3
Express $\frac{9 x ^{2} + 14 x - 1}{( 3 x - 1 ) ( x + 2 )}$ in partial fractions.
[5 marks]
Answer

Example 4

Express $\frac{x ^{3} - 2 x ^{2} + 4}{x ^{2} ( x - 2 )}$ in partial fractions.

Hint Same degree, and a repeated factor

x^{2}

: use

A + \frac{B}{x} + \frac{C}{x ^{2}} + \frac{D}{x - 2}

B4
Express $\frac{2 x ^{3} - x + 5}{( x - 1 ) ^{2} ( x + 2 )}$ in partial fractions.
[6 marks]
Answer

Practice

A-Level

C1
Raj says: " $\frac{x ^{2} + 4 x - 1}{( x - 1 ) ( x + 3 )} \equiv \frac{1}{x - 1} + \frac{1}{x + 3}$ ." Explain the mistake, give the correct partial fractions, and check your answer at $x = 0$ .
[5 marks]
Answer

C2
Write $\frac{x ^{3} + 3 x ^{2} - 2}{x + 2}$ in the form $a x^{2} + b x + c + \frac{d}{x + 2}$ .
[4 marks]
Answer
C3
Given that $\frac{x ^{3} + a x ^{2} + 2}{x ^{2} + x - 2} \equiv x + 2 + \frac{B}{x - 1} + \frac{C}{x + 2}$ , find $a$ , $B$ and $C$ .
[5 marks]
Answer $a = \dots$ , $B = \dots$ , $C = \dots$
C4
Lucy says that $\frac{3 x ^{2} + 1}{x ^{2} + x}$ is a proper fraction because the bottom has two terms and the top has two terms. Explain why she is wrong, then express the fraction in partial fractions.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Express $\frac{4 x ^{3} - 4 x + 10}{2 x ^{2} + x - 3}$ in partial fractions.
[7 marks]
Answer

D2
$f (x) = 2 x^{3} + 3 x^{2} - 3 x - 2$ .
(a)
Show that $(x - 1)$ is a factor of $f (x)$ and hence factorise $f (x)$ fully.
[3 marks]
Answer
(b)
Hence express $\frac{4 x ^{3} + 8 x ^{2} + 2 x - 5}{f ( x )}$ in partial fractions.
[6 marks]
Answer
Total for D2: 9 marks
D3
Zoe writes $\frac{x ^{3} + 1}{x ^{2} - x} = x + \frac{3}{x}$ .
(a)
By substituting $x = 2$ , show that Zoe's final answer is wrong.
[2 marks]
(b)
Express $\frac{x ^{3} + 1}{x ^{2} - x}$ correctly in partial fractions.
[6 marks]
Answer
Total for D3: 8 marks
D4
$f (x) = \frac{2 x ^{2} + 7 x + 7}{( x + 1 ) ( x + 2 )}$ , $x \geq 0$ .
(a)
Express $f (x)$ in the form $A + \frac{B}{x + 1} + \frac{C}{x + 2}$ .
[5 marks]
Answer $A = \dots$ , $B = \dots$ , $C = \dots$
(b)
Hence show that $\int_{0}^{2} f (x) d x = 4 + ln \frac{9}{2}$ .
[6 marks]
Total for D4: 11 marks
D5
$f (x) = \frac{2 x ^{2} + 5 x + 5}{x + 2}$ , $x \neq = - 2$ .
(a)
Write $f (x)$ in the form $a x + b + \frac{c}{x + 2}$ .
[3 marks]
Answer
(b)
Show that $f (x) > 2 x + 1$ for all $x > - 2$ .
[3 marks]
(c)
Solve $f (x) = x + 4$ . Give your answers in surd form.
[3 marks]
Answer
Total for D5: 9 marks
D6
Given that $\frac{4 x ^{3} + p x ^{2} + q x + 5}{( x - 1 ) ( x + 3 )} \equiv 4 x + 1 + \frac{3}{x - 1} + \frac{R}{x + 3}$ , find the values of $p$ , $q$ and $R$ .
[7 marks]
Answer $p = \dots$ , $q = \dots$ , $R = \dots$

Extension

Stretch

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

E1
Find all the integers $x$ for which $\frac{x ^{4} + 2 x ^{3} + 3}{x ^{2} + x - 2}$ is an integer.
[9 marks]
Hint 1 · What to try
Divide $x^{4} + 2 x^{3} + 3$ by $x^{2} + x - 2$ . The quotient is always an integer when $x$ is.
Hint 2 · The first line
The remainder is $x + 5$ . Split $\frac{x + 5}{( x - 1 ) ( x + 2 )}$ into partial fractions.
Hint 3 · The full method
When $x$ is large, $\frac{2}{x - 1} - \frac{1}{x + 2}$ is a small number that is not zero. Check the small integers one by one.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can tell whether an algebraic fraction is improper by comparing degrees.
I can divide polynomials to find the quotient and the remainder.
I can split an improper fraction into a polynomial part and partial fractions.
I can find unknown constants in an identity by comparing coefficients.

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