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 .

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.

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

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.

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 .

Expand and simplify .

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