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

One of the three expressions below is an improper algebraic fraction.

Identify the improper algebraic fraction and write it in the form 

  $Ax+B+\frac{C}{x+1}$

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

One of the three expressions below is an improper algebraic fraction.

Identify the improper algebraic fraction and write it in the form 

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

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

Simplify 

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

Solve the equation

$\frac{x^3-2x^2-8x}{x-4}=x^2+10x+16$

Simplify

 $\frac{x^2+x-2}{x^3+4x^2-4x-16}$

Simplify

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

Simplify

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

Given that

$\frac{x^3+5x^2-4}{x-5}\equiv A{x}^2+Bx+C+\frac{D}{x-5}x\in \mathrm{ℝ}x\ne 5$

find the values of the constants $A$, $B$, $C$ and $D$.

Given that

$\frac{x^3+3x^2-2x+4}{x+1}\equiv A{x}^2+Bx+C+\frac{D}{x+1}x\in \mathrm{ℝ}x\ne -1$

find the values of the constants $A$, $B$, $C$ and $D$.

Simplify 

$\frac{x^3-7x^2+14x-8}{x-1}$

Solve the equation

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

Given that

$\frac{x^2+ax-5}{x+1}\equiv x+3+\frac{d}{x+1}x\in \mathrm{ℝ}x\ne -1$

find the values of the constants $a$ and $d$.

Given that

$\frac{2x^3-4x+3}{x-2}\equiv P{x}^2+Qx+R+\frac{S}{x-2}x\in \mathrm{ℝ}x\ne 2$

find the values of the constants $P$, $Q$, $R$ and $S$.

Given

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

find $\mathrm{f}\left(x\right)$ in the form $a{x}^2+bx+c$, where $a$, $b$ and $c$ are constants to be found.

When 

$x^3+a{x}^2+4x-1$

is divided by $\left(x+2\right)$

• the quotient is $\left(x^2-4x+12\right)$

• the remainder is $b$

Find the values of $a$ and $b$.

Simplify 

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

Simplify 

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

Simplify

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

One of the three expressions below is an improper algebraic fraction.

Identify the improper algebraic fraction and write it in the form 

  $A{x}^2+Bx+C+\frac{D}{x-4}$

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

Given that

$\frac{3x^4+2x^3-5x+2}{x-3}\equiv A{x}^3+B{x}^2+Cx+D+\frac{E}{x-3}x\in \mathrm{ℝ}x\ne 3$

find the values of the constants $A$, $B$, $C$, $D$ and $E$.

Given

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

find the values of the constants $a$ and $b$.

When 

$2x^3+(a+b)x^2+(a-b)x-3$

is divided by $\left(x+4\right)$

• the quotient is $2x^2+(2a+3)x+(2b-5)$

• the remainder is $c$

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