Edexcel A-Level Mathematics A2 P4.2 Algebra and Functions Questions

Practise rewriting rational functions as partial fractions and solving for constants in forms used for later integration.

Exam points

express rational functions in the requested partial fraction form with constants A, B and C
compare coefficients or substitute roots to find constants in linear and repeated factors

Covered syllabus points

P4.2.1 - Decompose rational functions into Partial fractions

[Maximum number: 3]

Figure 2

Figure 2 shows a sketch of the curve defined by the parametric equations

x=t^{2}+2 t \quad y=\frac{2}{t(3-t)} \quad a \leqslant t \leqslant b

where a and b are constants.
The ends of the curve lie on the line with equation y=1

Write $\frac{t+1}{t(3-t)}$ in partial fractions.

(ii)

$\frac{t+1}{t(3-t)}=\frac{A}{t}+\frac{B}{3-t}$ $\Rightarrow t+1=A(3-t)+B t$

E.g. $t=0 \Rightarrow 1=3 A \Rightarrow A=\frac{1}{3} ; t=3 \Rightarrow 4=3 B \Rightarrow B=\frac{4}{3}$ $\frac{t+1}{t(3-t)}=\frac{1}{3 t}+\frac{4}{3(3-t)}$

[Maximum number: 3]

f(x)=\frac{5 x+10}{(1-x)(2+3 x)}

Write f(x) in partial fraction form.

$\frac{5 x+10}{(1-x)(2+3 x)} \equiv \frac{A}{1-x}+\frac{B}{2+3 x} \Rightarrow$ Value for A or B
One correct value, either A=3 or B=4
Correct PF form $\frac{3}{1-x}+\frac{4}{2+3 x}$
(3)

[Maximum number: 4]

f(x)=\frac{2 x^{4}+15 x^{3}+35 x^{2}+21 x-4}{(x+3)^{2}} \quad x \in \mathbb{R} \quad x>-3

Find the values of the constants A, B, C and D such that

\mathrm{f}(x)=A x^{2}+B x+C+\frac{D}{(x+3)^{2}}

Any correct constant, so A=2, B=3, C=-1, or D=5 $2x^4+15x^3+35x^2+21x-4=Ax^2(x+3)^2+Bx(x+3)^2+C(x+3)^2+D$ $\Rightarrow A=\ldots,\ B=\ldots,\ C=\ldots,\ D=\ldots$

or

$2x^4+15x^3+35x^2+21x-4\div(x^2+6x+9)=\ldots x^2+\ldots x+\ldots+\frac{\ldots}{(x+3)^2}$
2 correct of A=2, B=3, C=-1, D=5A=2, B=3, C=-1, D=5
(4)

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Covered syllabus points

Question 1

Question 2

Question 3