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

Let

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

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{3x^2-6x+2}{x^3-3x^2+2x}$

Express $\text{f}\left(x\right)$ in partial fractions.

(i)

Briefly explain why

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

is an improper (algebraic) fraction.

(ii)

Show that

$\left(x^2+x-4\right)÷\left(x^2+4x+3\right)=1-\frac{\left(3x+7\right)}{x^2+4x+3}$

Express

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

as partial fractions.

Hence, show that

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

Let

$\text{f}\left(x\right)=\frac{x^2+x+5}{\left(x+2\right)\left(3+x^2\right)}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{3\left(x^2-x+1\right)}{\left(x+1\right){\left(x-2\right)}^2}$

Express $\text{f}\left(x\right)$ in partial fractions.

It is given that

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

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

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

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

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

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

Express

$\frac{3x^2-13}{x^3-13x+12}$

in partial fractions.

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

Express $\frac{5x}{\left(x+2\right)\left(x-3\right)}$ in partial fractions.

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

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

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

Express

$\frac{19-8x}{x^3-3x^2+4}$

in partial fractions.

Let

$\text{f}\left(x\right)=\frac{3x+11}{x^2+6x+9}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{11x^2-2x+14}{\left(2x-1\right)\left(5+x^2\right)}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

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

Express $\text{f}\left(x\right)$ in partial fractions.

It is given that

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

(i) Show that $\text{f}\left(-3\right)=0$

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

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

Express

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

in partial fractions.

Let

$\text{f}\left(x\right)=\frac{2x^2+16x-24}{x^3-2x^2-8x}$

Express $\text{f}\left(x\right)$ in partial fractions.

It is given that

$\text{f}\left(x\right)=\frac{x^2+x+5}{x^2+3x+2}$

Show that

$\text{f}\left(x\right)\equiv 1-\frac{2x-3}{\left(x+1\right)\left(x+2\right)}$

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

Hence express $\text{f}\left(x\right)$ in partial fractions.

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

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

Express

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

in partial fractions.

Let

$\text{f}\left(x\right)=\frac{8}{x^3-6x^2+8x}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{13x^2-10}{x^3-x^2-2x}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{x^2-3}{\left(x-3\right){\left(x+2\right)}^2}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{x^2-5}{x^3-x^2-17x-15}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{3x^2-22x+25}{x^3-7x^2+8x+16}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{2x^2-3}{x^3+4x^2-3x-18}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{16x^2-7x+1}{3x^3-x^2}$

Express $\text{f}\left(x\right)$ in partial fractions.

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

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

Let

$\text{f}\left(x\right)=\frac{34-3x^2-x}{x^3-5x^2+3x+9}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{x^2+3x+10}{x^2+8x+15}$

Express $\text{f}\left(x\right)$ in partial fractions.

Let

$\text{f}\left(x\right)=\frac{2x-5}{x^3-3x^2-13x+15}$

Express $\text{f}\left(x\right)$ in partial fractions.