Polynomials (CIE A Level Maths: Pure 3)

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  

It is given that

Find .

The result of dividing 

Find the values of and .

Consider the function 

Find the quotient and remainder when is divided by  

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 

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  

Find the remainder when    is divided by 

  [2]

Find the value of when .

  [1]

[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.

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 .