The polynomial function is .

(a)

Find and .

Substitute at in the polynomial function.

.

.

Substitute at in the polynomial function.

.

and .

(b)

Intermediate value theorem :

The function is continuous on the closed interval , let be the number between  and  , where then exist a number in such that .

From (a) :

and .

Therefore .

Then according to intermediate value theorem, there exist atleast one root such that .

(c)

An - intrcepts ,The limits are , to four decimal places.

The polynomial function is and intervals .

Newtons approximation method formula : .

Differentiate on each side with respect to .

.

Newtons approximation method formula : .

Consider root as .

Newtons approximation : .

For .

Newtons approximation : .

.

For .

Newtons approximation : .

.

For .

Newtons approximation : .

Since  and are same to four decimal places, the root of equation .

The root of the equation .

(d)

Graph :

Graph  the polynomial function :

Observe the graph :

The polynomial touches -axis at .

The root of the equation is .

(e)

Graph :

Graph  the polynomial function :

Observe the graph :

satisfies the graph of the poynomial.

The root of the equation is .

(a)  and .

(b) There exist atleast one root in such that .

(c) The root of the equation is ..

(d)The root of the equation is .

(e) satisfies the graph of the poynomial.