The function is .

Use the mean value theorem to show that the function has atleast one real root.

Mean value Theorem :

If  is continuous on  and differentiable on open interval , then there exists a number  in .

The function  is continuous and differentiable, since it is a polynomial function.

So the function have atleast one real root on open interval .

Use the Rolles theorem to show that the function has only one real root.

Rolles Theorem :

Let  be a function that satisfies the following three hypotheses.

1.  is continuous on .

2.  is differentiable on .

3. .

Then there is a number  in  such that .

The function  is continuous and differentiable, since it is a polynomial function.

Suppose ,  are distinct real numbers such that .

Apply derivative with respect to .

From the Rolles theorem number  in  such that .

Therefore the function does not have the real roots while equting .

From Rolles theorem, above four root are imaginary.

But according to Mean value theorem the remaining one root is real.

It is clear that the functon have only real root.

The function  has exactly one real solution.