2.7 Polynomial Functions

Let a function  be defined by . 

(i)

Show that  is a factor of .

(ii)

Hence factorise fully.

(iii)

Write down all the solutions to

Consider the polynomial 

Show that 2 is a root of .

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:  

Given that  is one of the roots of the equation  find the other two roots.

For the function  ,  the sum of the roots is and the product of the roots is .  Find the values of and .

The function has three real and two complex roots. 

It is given for that the sum of the roots is and the product of the roots is . 

 and are non-real roots of the equation ,  where  is a constant. 

Find and , in terms of .

Consider the function ,  where  is a constant.

It is given that  is a factor of . 

Find the value of .

Consider the function

where  for . 

The graph of , shown below, passes through .  The roots of  are  and .

Explain why  must be even.

A polynomial function  is defined by where  and are positive constants with .

Sketch the graph of . Label the coordinates where the graph crosses the coordinate axes.

Consider the function , where  and  are positive constants. The points  and  lie on the graph .

Find  and  in terms of  and .

Consider the function  defined by ,  where  are constants.

Given that  is a factor of , and that the sum of the roots of the equation  is 5, 

(i)

find the values of  and , and

(ii)

hence factorise  fully.

Consider the function  defined by  ,  where and  are real constants. 

It is given that the sum of the roots of the equation   is  ,  and that the product of the roots is . 

Find a set of values for and  that satisfies the above conditions, such that . .

The equation  has non-real roots  and  where  . 

The equation  has roots  and .

Find the values of and .

Consider the polynomial function defined by

Where the are real constants. The function has the property that  for all values of . 

Show that 

Consider the polynomial function ,  where . Two distinct roots of  are given by  and ,  where  is a real constant. The remainder when is divided by  is 8100.

(i)

Find the two possible values of .

 

(ii)

Hence find real values for and  such that  is guaranteed to be a factor of .

The polynomial function  is defined by

where  is a real constant. 

The graph of  only intersects the -axis at the point .

By considering the sum of the roots, use proof by contradiction to show that  has two non-real roots.