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

The function is defined by

Show that

where and are constants to be found.

Given that

find the values of the constants , and

Hence, using algebraic integration, find the exact value of

giving your answer in the form where and are integers to be found.

Simplify

(i)  

(ii)  

(iii)  

Simplify fully 

Simplify fully 

Simplify fully 

The function  is given by

Show that .

Hence write down a factor of .

Fully factorise .

Write down the solutions to the equation .

Show that is a factor of .

Fully factorise .

Find all the real solutions to .

Given that is a factor of  find the value of .

Work out .

Work out  .

Given  

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

(i) Write down the degree of . 

(ii) Write down the degree of . 

(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  , where  and  are integers to be found.

Simplify fully 

Hence solve the equation .

It is given that

Find .

The result of dividing  by  is .

Find the values of  and .

Simplify fully

(i)  

(ii)  

(iii)  

Simplify fully 

Simplify fully 

Simplify fully 

The function  is given by

Show that  is a factor of .

Hence, or otherwise, fully factorise .

Write down the roots of .

Show that is a factor of .

Hence find all the real solutions to the equation .

Given that  is a factor of  find the value of .

Hence, or otherwise, fully factorise .

Work out .

Work out  .

(i) Find the remainder when  is divided by .

(ii) Find the value of  when .

(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  , where  and  are integers to be found.

Simplify 

Hence solve .

It is given that

Why would assuming that  be a logical first step in attempting to determine the precise forms of  and ?

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

Explain, with an example, why the forms of  and  determined in parts (a) and (b) are not the only possible forms for those functions.

When  is divided by  the quotient is  and the remainder is .

Find the values of  and .

Simplify fully

(i)  

(ii)  

(iii)  

Simplify fully 

Simplify fully 

Simplify fully 

Given that  is a factor of the function

find the value of  and fully factorise .

Show that is a factor of  and hence find all the real solutions to the equation .

Show that  is a factor of .

Given that  is a root of  , find the value of .

Work out .

Work out 

For a polynomial , the Remainder Theorem states that

When  is divided by  the remainder is .

Use the Remainder Theorem to find the remainder when is divided by .

Work out the remainder when  is divided by .

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 

It is given that

where  and  are integers.

Find the values of .

When is divided by  the quotient is and the remainder is .

Find the values of and .