Polynomials (Edexcel A Level Maths: Pure): Exam Questions

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

Find the constants , and such that

Hence show that has only one real root.

Write down the real root of the equation

where is a constant

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

A function is given by

Given that  is a factor, use an algebraic method to factorise .

Give your answer in the form

where and are integers to be found.

Sketch the curve with equation , labelling the coordinates of any points at which the curve meets the coordinate axes.

A function is given by

Given that  is a root of the equation , find the possible values of .

A function is given by

Given that  is a solution to the equation , use algebra to factorise as far as possible.

A function is given by 

The equation has a solution at .

Use algebra to factorise into three linear factors.

Show that

where , , and are constants to be found.

Use polynomial division to divide  by .

A function is defined by

Given that  is a factor, factorise completely.

A function is defined by

Show that

where , and are constants to be found.

Hence factorise into three linear factors.

Write down all real roots to the equation .

A function is given by

Show that is a factor of .

Factorise  completely.

Solve .

Given that  is a factor of , factorise  completely.

Show that

where , , and are constants to be found.

Show that where  and  are constants to be found.

Hence factorise  completely.

Solve .

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

Factorise  completely.

Solve .

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

where is a positive constant.

Given is a factor of , show that

Hence

(i) Find the value of

(ii) use algebra to find the exact solutions of the equation

(i) Calculate

(ii) Write as a product of two algebraic factors.

Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation

The function  is given by

where  and  are constants.

Given that both and  are factors of , find the values of and .

A function is defined as

where and are constants.

Given that

find the values of  and .

Factorise completely.

Given that

find the values of , , and .

Factorise completely .

A square has a side length of  units.

Find an expression for the length of the diagonal of the square, in terms of  and .

Give your answer in the form

where , , , , and are constants to be found.

Given that

find the values of  and .

Factorise  completely.

Given that is a root of the equation

show that the equation has no other real roots.

Figure 4 shows a sketch of part of the curve with equation

and part of the curve with equation

Verify that the curves intersect at

The curves intersect again at the point

Using algebra and showing all stages of working, find the exact coordinate of

In this question you must show detailed reasoning.

Solutions relying on calculator technology are not acceptable.

The curve has equation

The curve has equation

Verify that when the curves and intersect.

The curves also intersect when .

Given that

use algebra to find the exact value of .

Given that

find the values of , , and .

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.

A function is defined as

Given that  is a factor of , use algebra to express  as the product of four linear factors.

Expand and simplify

A cuboid has a length of  units, a width of  units, and a height of units. 

Expand and simplify an expression for the volume of the cuboid in terms of  and .

Find and .

Solve