Partial Fractions (Edexcel A Level Maths: Pure): Exam Questions

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

Express

$\frac{6x-13}{x^2-5x+6}$

in partial fractions.

Express  $\frac{7x+34}{x^2+9x+14}$  in partial fractions.

Express  $\frac{12}{x^2+4x-5}$  in partial fractions.

Express  $\frac{2x}{x^2-9x-52}$  in partial fractions.

Express

$\frac{3x+8}{{\left(x+1\right)}^2}$

in the form

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

where $A$and $B$ are integers to be found.

Express  $\frac{3x+11}{x^2+6x+9}$  in partial fractions.

Express

$\frac{3x+7}{x^2+4x+3}$

in partial fractions.

Hence, show that

$\frac{x^2+x-4}{x^2+4x+3}\equiv K-\frac{2}{x+1}-\frac{1}{x+3}$

where $K$ is a constant to be found.

Write

$\frac{12x-6}{\left(x-3\right)\left(x+2\right)\left(x-1\right)}$

in the form 

$\frac{A}{x-3}+\frac{B}{x+2}+\frac{C}{x-1}$

 where $A$, $B$ and $C$ are integers to be found.

Show that $(x-2)$ is a factor of  $x^3+4x^2-3x-18$.

Factorise fully $x^3+4x^2-3x-18$.

Express 

$\frac{7x^2+32x+8}{x^3+4x^2-3x-18}$

in the form

$\frac{A}{x-2}+\frac{B}{x+p}+\frac{C}{{\left(x+p\right)}^2}$

where $A$, $B$, $C$ and $p$ are integers to be found.

A function is given by

$\mathrm{f}\left(x\right)=x^3-13x+12$

(i) Show that $\mathrm{f}\left(1\right)=0$.

(ii) Hence write down a factor of $\mathrm{f}\left(x\right)$.

(i) Show that $\mathrm{f}\left(3\right)=0.$

(ii) Hence write down a second factor of $\mathrm{f}\left(x\right)$.

Fully factorise $\mathrm{f}\left(x\right)$.

Express  $\frac{3x^2-13}{x^3-13x+12}$  in partial fractions.

A function is given by

$\mathrm{f}\left(x\right)=x^3+3x^2-4x-12$

Find $\mathrm{f}(-3)$.

Factorise $\mathrm{f}\left(x\right)$.

Give your answer in the form

$\mathrm{f}\left(x\right)=\left(x+p\right)\left(x+q\right)\left(x-q\right)$

where $p$ and $q$ are integers to be found.

Hence, express

$\frac{3x^2-8x-36}{x^3+3x^2-4x-12}$

in partial fractions.

$\frac{1+11x-6x^2}{\left(x-3\right)\left(1-2x\right)}\equiv A+\frac{B}{\left(x-3\right)}+\frac{C}{\left(1-2x\right)}$

Find the values of the constants $A$, $B$ and $C$.

$\mathrm{f}\left(x\right)=\frac{50x^2+38x+9}{{\left(5x+2\right)}^2\left(1-2x\right)}x\ne -\frac{2}{5}x\ne \frac{1}{2}$

Given that $\mathrm{f}\left(x\right)$ can be expression in the form

$\frac{A}{5x+2}+\frac{B}{{\left(5x+2\right)}^2}+\frac{C}{1-2x}$

where $A$, $B$ and $C$ are constants,

(i) find the value of $B$ and the value of $C$,

(ii) show that $A=0$.

Write

 $\frac{x^2+x+5}{x^2+3x+2}$

in the form

$A-$$\frac{2x-3}{\left(x+1\right)\left(x+2\right)}$

where $A$ is an integer to be found.

Express $\frac{2x-3}{\left(x+1\right)\left(x+2\right)}$  in the form  $\frac{B}{x+1}+\frac{C}{x+2}$  where $B$and $C$ are integers to be found.

Hence, write

$\frac{x^2+x+5}{x^2+3x+2}$

  in the form

 $P+\frac{Q}{x+1}+\frac{R}{x+2}$

where $P$, $Q$ and $R$ are integers to be found.

Write

$(x^2+4x-6)÷(x^2-x-6)$

in the form

$P+\frac{Qx}{x^2-x-6}$

where $P$ and $Q$ are integers to be found.

Using your value of $Q$ from part (a), write

$\frac{Qx}{\left(x+2\right)\left(x-3\right)}$

  in the form 

$\frac{A}{x+2}+\frac{B}{x-3}$

Hence, write

$\frac{x^2+4x-6}{x^2-x-6}$

as the sum of a constant and two partial fractions.

Show that $(x+1)$ is a factor of  $x^3-3x^2+4$.

Factorise fully $x^3-3x^2+4$.

Hence, express

$\frac{19-8x}{x^3-3x^2+4}$ 

in partial fractions.

Write

$\frac{3x^2-6x+2}{x^3-3x^2+2x}$

in the form

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

where $A$, $B$ and $C$ are integers to be found.

Express

$\frac{5x^2+10x+8}{x{\left(x+2\right)}^2}$

in partial fractions.

Express  $\frac{2x^2+16x-24}{x^3-2x^2-8x}$  in partial fractions.

Express $\frac{8}{x^3-6x^2+8x}$ in partial fractions.

Write

$\frac{13x^2-10}{x^3-x^2-2x}$ 

in the form 

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

where $A$, $B$ and $C$ are integers to be found.

Write 

$\frac{x^2+3x+10}{x^2+8x+15}$

in the form 

$A+\frac{B}{x+3}+\frac{C}{x+5}$

where $A,B$and $C$ are integers to be found.

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

A function is given by

$\mathrm{f}\left(x\right)=x^3+3x^2-4x-12$

By finding $\mathrm{f}(-3)$, or otherwise, fully factorise $\mathrm{f}\left(x\right)$.

Express

$\frac{3x^2-8x-36}{\mathrm{f}\left(x\right)}$

in partial fractions.

$\mathrm{f}\left(x\right)=\frac{3kx-18}{\left(x+4\right)\left(x-2\right)}$ where $k$ is a positive constant

Express $\mathrm{f}\left(x\right)$ in partial fractions in terms of $k$.

Express

$\frac{3x^2-22x+25}{x^3-7x^2+8x+16}$

in partial fractions.

Express

$\frac{x^2-5}{x^3-x^2-17x-15}$ 

in partial fractions.

Write

$\frac{34-3x^2-x}{x^3-5x^2+3x+9}$

in the form 

$\frac{A}{x+1}+\frac{B}{x-k}+\frac{C}{{\left(x-k\right)}^2}$ 

where $A$, $B$, $C$ and $k$ are integers to be found.

Express 

$\frac{2x^2-3}{x^3+4x^2-3x-18}$ 

in partial fractions.

Express 

$\frac{2x-5}{x^3-3x^2-13x+15}$ 

in partial fractions.

Express 

$\frac{2\left(3x^3+x^2-17x+4\right)}{x\left(x-1\right)\left(x-4\right)\left(x+2\right)}$ 

in partial fractions.

Given that 

$x^4-4x^3-2x^2+12x+9\equiv (x^2-6x+9)(x^2+2x+1)$

express 

$\frac{5x^3-15x^2+19x+7}{x^4-4x^3-2x^2+12x+9}$

in partial fractions.