Polynomials (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

Expand and simplify .

A rectangle has side lengths of  units and  units. Find an expression for the area of the rectangle in terms of  and .

Given that , where  are constants, find the values of .

Factorise completely .

Divide  by .

Find the remainder when is divided by .

Given that  is a factor of , factorise completely.

Show that  where  are constants to be found.

Hence factorise completely.

Write down all the real roots of the equation .

Use the factor theorem to show that is a factor of .

Factorise  completely.

Write down all the real roots of the equation .

. Given that and :

find the values of  and .

Factorise  completely.

The function  is given by

Show that 

Hence write down a factor of 

Fully factorise .

Write down the solutions to the equation 

Show thatis a factor of .

Fully factorise 

Find all the real solutions to 

Given that is a factor of   find the value of .

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.

It is given that

Find .

The result of dividing 

Find the values of and .

Consider the function 

(i) Find the quotient and remainder when is divided by  

(ii) Hence write   in the form where  and  are constants to be determined.

The function   has  as a factor, and when  is divided by the remainder is 7. 

Show that and  must satisfy the simultaneous equations:  

Hence find and 

Show that  is a factor of .

Fully factorise .

Sketch the graph of .

Label any points where the graph crosses the coordinate axes.

Expand and simplify .

A square has side lengths of  units. Find an expression for the length of the diagonal of the square in terms of  and .

Given that , where  are constants, find the values of  and .

Factorise completely .

Divide  by .

Find the remainder when is divided by .

Given that  is a factor of , factorise  completely.

Show that where  and  are constants to be found.

Hence factorise  completely.

Write down all the real roots of the equation .

Use the factor theorem to show that  is a factor of .

Factorise  completely.

Write down all the real roots of the equation .

. Given that  and : 

find the values of  and .

Factorise  completely.

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  

(i) Find the remainder when    is divided by 

  [2]

(ii) Find the value of when .

  [1]

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

[1]

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 .

Given that is a factor of the function  and that the remainder when  is divided by  is ,  find the values of the constants p and q.

Show that can be written in the form  where  and  are constants to be found.

Given that  is a factor of , sketch the graph of . Label any points where the graph intersects the coordinate axes.

(There is no need to label any stationary points.)

Show that  is a factor of the function  and hence, or otherwise, fully factorise .

Sketch the graph of .

Label any points where the graph crosses the coordinate axes.

Find the coordinates of the points of intersection between the curve with equation  and the line with equation .

On the same diagram, sketch the graphs of  and .

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

Expand and simplify .

A cuboid has a length of  units, a width of  units, and a height of .  Find an expression for the volume of the cuboid in terms of  and .

Given that , where  and are constants, find the values of  and .

Factorise completely .

Divide  by .

Find the remainder when  is divided by .

Given that  is a factor of , factorise  completely.

Show that  where  and  are constants to be found.

Given that  is a factor of , factorise  completely.

Hence show that the equation  has exactly 2 real roots.

Given that 3 is a root of the equation , prove that the equation has no other real roots.

Show that  and .

Hence, solve .

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 .

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 .

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

Find the values of and .

 are factors of , where .

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.  (There is no need to label any stationary points).

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.

Fully factorise .

Sketch the graph of .

Label any points where the graph crosses the coordinate axes.

On the same diagram, sketch the graphs of and .

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

On the same diagram, sketch the graphs of  and .

Write down the number of solutions to the equation

.