Partial Fractions (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Express  $\frac{2}{x\left(x+2\right)}$  as partial fractions.

Write  $\frac{x-10}{\left(x+2\right)\left(x-1\right)}$  in the form  $\frac{A}{x+2}+\frac{B}{x-1}$ , where $A$and $B$ are integers to be found.

It is given that

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

(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 another factor of $\mathrm{f}\left(x\right)$.

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

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

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

Use algebraic division to work out $(x^2+4x-6)÷(x^2-x-6)$.

Write $\frac{5x}{\left(x+2\right)\left(x-3\right)}$  in the form $\frac{A}{x+2}+\frac{B}{x-3}$

Hence, or otherwise, show that $\frac{x^2+4x-6}{x^2-x-6}=1+\frac{2}{x+2}+\frac{3}{x-3}$

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

Use the factor theorem to show that $(x+1)$ is a factor of  $x^3-3x^2+4$.

Hence, or otherwise, fully factorise $x^3-3x^2+4$.

Show that $\frac{19-8x}{x^3-3x^2+4}$  can be written in the form $\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{{\left(x-2\right)}^2}$  , where $A,B$and $C$ are integers to be found.

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

Express  $\frac{x^2-5}{x^3-x^2-17x-15}$  as partial fractions.

Express  $\frac{11x^2-2x+14}{\left(2x-1\right)\left(x^2+5\right)}$  as partial fractions.

Express $\frac{4-4x-4x^2}{(3x-1)(x^2-2x+1)}$ as partial fractions.

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

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.

It is given that

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

(i) Show that $\mathrm{f}(-3)=0$

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

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

Express  $\frac{3x^2-8x-36}{x^3+3x^2-4x-12}$ as partial fractions.

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

Show that   $\frac{x^2+x+5}{x^2+3x+2}$ can be written 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.

Using your values for $A,B$and $C$ write $\frac{x^2+x+5}{x^2+3x+2}$  in the form $A+\frac{B}{x+1}+\frac{C}{x+2}$

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

Use the factor theorem to show that $(x-2)$ is a factor of  $x^3+4x^2-3x-18$.

Hence, or otherwise, fully factorise $x^3+4x^2-3x-18$.

Show that  $\frac{7x^2+32x+8}{x^3+4x^2-3x-18}$ can be written in the form  $\frac{A}{x-2}+\frac{B}{x+3}+\frac{C}{{\left(x+3\right)}^2}$ , where $A,B$and $C$ are integers to be found.

Express  $\frac{3x^2-22x+25}{x^3-7x^2+8x+16}$  as partial fractions.

Express  $\frac{2x^2-3}{x^3+4x^2-3x-18}$  as partial fractions.

Express  $\frac{16x^2-7x+1}{3x^3-x^2}$  as partial fractions.

Given that  $\mathrm{f}\left(\frac{1}{4}\right)=0,$ fully factorise $\mathrm{f}\left(x\right)$  where $\mathrm{f}\left(x\right)=8x^3-2x^2-4x+1.$

Hence express  $\frac{5x^2-2x-2}{\mathrm{f}\left(x\right)}$  as partial fractions.

Express  $\frac{8}{x^3-6x^2+8x}$  as 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.

Show that  $\frac{34-3x^2-x}{x^3-5x^2+3x+9}$  can be written in the form  $\frac{A}{x+1}+\frac{B}{x-3}+\frac{C}{{\left(x-3\right)}^2}$ , where $A,B$and $C$ are integers to be found.

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

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.

It is given that

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

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

(ii) Hence, fully factorise $\mathrm{f}\left(x\right)$ .

Express $\frac{3x^2-8x-36}{\mathrm{f}\left(x\right)}$ as partial fractions.

Express  $\frac{2x-5}{x^3-3x^2-13x+15}$  as partial fractions.

Express  $\frac{2\left(3x^3+x^2-17x+4\right)}{x\left(x-1\right)\left(x-4\right)\left(x+2\right)}$  as partial fractions.

Given that  $x^4-4x^3-2x^2+12x+9=(x^2-6x+9)(x^2+2x+1)$, express $\frac{5x^3-15x^2+19x+7}{x^4-4x^3-2x^2+12x+9}$ as partial fractions.

Express  $\frac{x^3+6x^2+45x}{x^4-81}$as partial fractions.

Show that  $\left(2x^2+5\right)\left(x^2+3x+2\right)=2x^4+6x^3+9x^2+15x+10$

Express  $\frac{13x^3+12x^2+28x+8}{2x^4+6x^3+9x^2+15x+10}$ as partial fractions.