Rational Expressions (Edexcel IAL Maths): Exam Questions

Simplify

(i) $\frac{x^{2}}{x}$

(ii) $\frac{x ( x - 1 )}{x}$

(iii) $\frac{6x+2}{2}$

(i) Factorise $x^{2}+7x+12$

(ii) Hence simplify $\frac{x^{2}+7x+12}{2(x+3)}$

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

Simplify fully $\frac{3x^{2}}{x+4}\times\frac{x^{2}+5x+4}{x}$

(i) Fully factorise $x^{3}-9x^{2}+20x$

(ii) Hence simplify $\frac{x^{3}-9x^{2}+20x}{x^{2}-5x}$

The function is given by $f(x)=2x^{3}+7x^{2}-4x$.

Show that $f(x)=x(2x-1)(x+4)$.

Hence, or otherwise, write down the real solutions to the equation

$\frac{f(x)}{x+1}=0.$

The function $f(x)$ is given by

$f(x)=x^{3}-4x^{2}-7x+10$

Work out $f(1)$ and hence write down a factor of $f(x)$.

Work out $f(x)\div(x+2)$.

Write $f(x)$ in the form $(x+a)(x+b)(x+c)$ where $a,b\mathrm{and}c$ are integers to be found.

Which one of the following algebraic fractions is improper? Explain your answer.

Find the remainder when $x^{3}+2x^{2}-5x+8$ is divided by $(x-3)$.

Given that $(x^{2}-8x-20)\div(x-2)=Ax+B+\frac{C}{x-2}$

where $A,B\mathrm{and}C$ are integer constants.

In terms of $A,B$ and/or $C$ as appropriate

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

Find the values of $A,B\mathrm{and}C$.

The function $f(x)$ is given by

$f(x)=x^{2}+ax+b$

where $a\mathrm{and}b$ are integer constants.

It is also given that $f(3)=f(-8)=0$.

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

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 $\frac{x^{2}+4x}{3x+6}\div\frac{2x+8}{x+2}$

The function $f(x)$ is given by

$f(x)=3x^{3}-5x^{2}-4x+4$

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

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

Fully factorise $f(x)$.

Write down the solutions to the equation $f(x)=0$.

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 $(x^{3}+3x^{2}-2x+4)\div(x+1)$.

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

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

(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

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

Find $f(x)$.

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

The function $f(x)$ is given by

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

Show that $(4x-3)$ is a factor of $f(x)$.

Hence, or otherwise, fully factorise $f(x)$.

Write down the roots of $f(x)$.

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 $(x^{3}+5x^{2}-4)\div(x-5)$.

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

(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

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

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

By first making the assumption from part (a), find $f(x)$.

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

When $x^{3}+ax^{2}+4x-1$ 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) $\frac{(x+2)^{2}+(x+2)(x+4)}{x^{2}+5x+6}$

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

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

$f(x)=2x^{3}+kx^{2}-11x-60$

find the value of $k$ and fully factorise $f(x)$.

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 $3ax^{2}+(a-6)x-2$.

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

Work out $(3x^{4}+2x^{3}-5x+2)\div(x-3)$.

Work out $\frac{x^{5}}{3x^{2}-2}$

For a polynomial $f(x)$, the Remainder Theorem states that

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

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 $c$.