Rational Expressions (OCR A Level Maths A: Pure): Exam Questions

Simplify

(i) $\frac{x}{2x}$  

(ii) $\frac{x+1}{x(x+1)}$  

(iii) $\frac{6x+12}{x^2+2x}$ 

Simplify fully $\frac{2x^2+6x}{x^3+3x^2}$

Simplify fully $\frac{x+4}{x^3}\times \frac{x^2+2x}{x+4}$

Simplify fully $\frac{x^2+4x}{3x+6}÷\frac{2x+8}{x+2}$

The function $\mathrm{f}\left(x\right)$ is given by

$\mathrm{f}\left(x\right)=3x^3-5x^2-4x+4$

Show that $\mathrm{f}\left(\frac{2}{3}\right)=0$.

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

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

Write down the solutions to the equation $\mathrm{f}\left(x\right)=0$.

Show that $(2x-3)$is a factor of $2x^3-13x^2+23x-12$.

Fully factorise $2x^3-13x^2+23x-12$.

Find all the real solutions to $2x^3-13x^2+23x-12=0$.

Given that $(2x-1)$ is a factor of $2x^3+x^2-25x+a$ find the value of $a$.

Work out $(x^3+3x^2-2x+4)÷(x+1)$.

Work out  $\frac{2x^3-4x+3}{x-2}$.

Given  $(x^2+8x-4)÷(x-3)=x+11+\frac{29}{x-3}$

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

(i) Write down the degree of $x^2+8x-4$. 

(ii) Write down the degree of $x-3$. 

(iii) Explain why you would expect the quotient to be of degree 1 in this case. 

One of the three algebraic fractions below is improper (‘top-heavy’).

Identify which fraction is improper and write it in the form  $A+\frac{B}{x+2}$, where $A$ and $B$ are integers to be found.

Simplify fully $\frac{x^3-2x^2-8x}{x-4}$

Hence solve the equation $\frac{x^3-2x^2-8x}{x-4}=x^2+10x+16$.

It is given that

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

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

The result of dividing $x^2+ax-5$ by $(x+1)$ is $x+3+\frac{d}{x+1}$.

Find the values of $a$ and $d$.

Simplify fully

(i) $\frac{x+3}{x^2+3x}$  

(ii) $\frac{x^3+x}{x^4}$  

(iii) $\frac{x^3+3x^2-4x}{x^4-x^3}$ 

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

Simplify fully $\frac{3x+9}{x+2}\times \frac{x^2+6x+8}{x+3}$

Simplify fully $\frac{x^2+8x-9}{x^2+7x+12}÷\frac{x^2+11x+18}{2x^2+7x-4}$

The function $\mathrm{f}\left(x\right)$ is given by

$\mathrm{f}\left(x\right)=4x^3-7x^2-21x+18$

Show that $(4x-3)$ is a factor of $\mathrm{f}\left(x\right)$.

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

Write down the roots of $\mathrm{f}\left(x\right)$.

Show that $(5x-2)$ is a factor of $25x^3+55x^2-56x+12$.

Hence find all the real solutions to the equation $25x^3+55x^2-56x+12=0$.

Given that $(4x-5)$ is a factor of $4x^3-9x^2+ax+30$ find the value of $a$.

Hence, or otherwise, fully factorise $4x^3-9x^2+ax+30$.

Work out $(x^3+5x^2-4)÷(x-5)$.

Work out  $\frac{3x^3+2x-5}{x^2+1}$.

(i) Find the remainder when $x^3-2x^2+4x-3$ is divided by $x-2$.

(ii) Find the value of $\mathrm{f}\left(2\right)$ when $\mathrm{f}\left(x\right)=x^3-2x^2+4x-3$.

(iii) Comment on your answers to parts (i) and (ii).  

One of the three algebraic fractions below is improper (‘top-heavy’):

Identify which fraction is improper and write it in the form  $Ax+B+\frac{C}{x+1}$, where $A,B$ and $C$ are integers to be found.

Simplify $\frac{x^3-7x^2+14x-8}{x-1}$

Hence solve $\frac{x^3-7x^2+14x-8}{x-1}=2x^2-5x+2$.

It is given that

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

Why would assuming that $\mathrm{g}\left(x\right)=x+1$ be a logical first step in attempting to determine the precise forms of $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$?

By first making the assumption from part (a), find $\mathrm{f}\left(x\right)$.

Explain, with an example, why the forms of $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ determined in parts (a) and (b) are not the only possible forms for those functions.

When $x^3+a{x}^2+4x-1$ is divided by $x+2$ the quotient is $x^2-4x+12$ and the remainder is $b$.

Find the values of $a$ and $b$.

Simplify fully

(i) $\frac{x^2+5x}{x^3+3x^2-10x}$  

(ii) $\frac{x^2-4}{x^4-16}$  

(iii) $\frac{(x+2{)}^2+(x+2\left)\right(x+4)}{x^2+5x+6}$ 

Simplify fully $\frac{x^2-5x+4}{x^3-2x^2-11x+12}$

Simplify fully $\frac{2x^2-x-6}{x^2-4}\times \frac{5x+10}{4x^2-9}$

Simplify fully $\frac{45x^3+90x^2-5x-10}{x^2-25}÷\frac{3x^2+7x+2}{2x^2+9x-5}$

Given that $(2x-5)$ is a factor of the function

$\mathrm{f}\left(x\right)=2x^3+k{x}^2-11x-60$

find the value of $k$ and fully factorise $\mathrm{f}\left(x\right)$.

Show that $(9x^2-4)$ is a factor of $9x^4-40x^2+16$ and hence find all the real solutions to the equation $9x^4-40x^2+16=0$.

Show that $(ax-2)$ is a factor of $3a{x}^2+(a-6)x-2$.

Given that $x=\frac{-1}{a-4}$ is a root of  $3a{x}^2+(a-6)x-2$, find the value of $a$.

Work out $(3x^4+2x^3-5x+2)÷(x-3)$.

Work out $\frac{x^5}{3x^2-2}$

For a polynomial $\mathrm{f}\left(x\right)$, the Remainder Theorem states that

When $\mathrm{f}\left(x\right)$ is divided by $(ax-b)$ the remainder is $\mathrm{f}\left(\frac{b}{a}\right)$.

Use the Remainder Theorem to find the remainder when $8x^3+6x^2-x-2$ is divided by $(2x+1)$.

Work out the remainder when $6x^2-x-2$ is divided by $(2x+1)$.

One of the three algebraic fractions below is improper (‘top-heavy’):

Identify which fraction is improper and rewrite it as a quotient and a remainder term.

Solve $\frac{8x^4+25x^3+3x^2-32x-4}{x^2+x-2}=2x^2+16x+3$

It is given that

$\frac{(x+a{)}^3}{x+b}=(x^2+7x+13)+\frac{1}{x+b}$

where $a$ and $b$ are integers.

Find the values of $a\mathrm{and}b$.

When $2x^3+(a+b)x^2+(a-b)x-3$ is divided by $x+4$ the quotient is $2x^2+(2a+3)x+(2b-5)$ and the remainder is $c$.

Find the values of $a,b$ and $c$.