Polynomial Functions (DP IB Maths: AA HL)

Consider the function ,  where  and  are constants.  It is given that  is a factor of .

Find the values of  and .

is a zero of the function where  is a constant.

As well as finding the value of , find all the solutions to the equation .

The point  is a turning point on the graph .

Given that   has three distinct real solutions, where  is a real constant, find the set of possible values of .

The graph of  is shown below, where is a polynomial function. The graph passes through the points and .

Given that the degree of is as small as possible, find an equation for .

The graph is translated by the vector  to form the graph , where  is a constant and  is a polynomial.

Given that is a factor of , find the possible values of .

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.

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

 and are non-real solutions of the equation . 
Given that   and ,  find the value of .

The function  has two integer solutions, one of which is double the other one. 

Find the value of .

Consider the function , where and  are real constants.

Show that if is a zero of then  is also a zero.

Let  be a polynomial defined by

(i)

is a factor of ,

(ii)

has exactly one real root.

Consider the function  defined by , where  is a real constant.

Given that the equation   has exactly three real roots, find the set of possible values of .

Consider the function , where  and  are constants.  It is given that  is a factor of .

Show that  and 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.