Partial Fractions (Edexcel IAL Maths): Exam Questions

Express

$\frac{1}{x ( x + 1 )}$

as partial fractions.

(i) Factorise $x^{2}+5x-6$.

(ii) Hence, or otherwise, express

$\frac{(5x+16)}{x^{2}+5x-6}$

as partial fractions.

Express

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

as partial fractions.

Write

$\frac{2(x-11)}{x^{2}+2x-15}$

in the form

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

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

Show that

$\frac{3 x + 8}{( x + 1 )^{2}} \equiv \frac{A}{x + 1} + \frac{B}{( x + 1 )^{2}}$

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

Express

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

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.

(i) Briefly explain why

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

is an improper (algebraic) fraction.

(ii) Show that

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

Express

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

as partial fractions.

Hence, show that

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

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

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

It is given that

$f(x)=x^{3}-13x+12.$

(i) Show that $f(1)=0$.

(ii) Hence write down a factor of $f(x)$.

(i)

Show that $f(3)=0.$

(ii)

Hence write down another factor of $f(x)$.

Fully factorise $f(x)$.

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 $(x^{2}+4x-6)\div(x^{2}-x-6)$.

Write $\frac{5 x}{( x + 2 ) ( x - 3 )}$ 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 $\frac{2 x}{( x + 1 ) ( x - 2 )}$ 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 $\frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{( x - 2 )^{2}}$ , 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 $\frac{12}{x^{2}+4x-5}$ as partial fractions.

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

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

(i) Show that $f(-3)=0$

(ii) Hence write down a factor of $f(x)$.

Fully factorise $f(x)$.

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 $A-\quad \frac{2 x - 3}{( x + 1 ) ( x + 2 )}$ , where $A$ is an integer to be found.

Express $\frac{2 x - 3}{( x + 1 ) ( x + 2 )}$ 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 $\frac{A}{x - 2} + \frac{B}{x + 3} + \frac{C}{( x + 3 )^{2}}$ , 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 $\frac{8}{x^{3}-6x^{2}+8x}$ 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 $\frac{A}{x + 1} + \frac{B}{x - 3} + \frac{C}{( x - 3 )^{2}}$ , where $A,B$and $C$ are integers to be found.

Express $\frac{x^{2} - 3}{( x - 3 ) ( x + 2 )^{2}}$ 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

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

(i) Show that $f(-3)=0$.

(ii) Hence, fully factorise $f(x)$ .

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

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

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