# Thread: Show an equation has at most a certain amount of roots

1. ## Show an equation has at most a certain amount of roots

Dont know how to answer the more general forms of these questions using Rolle's Theorem and mean value theorem, for example;

- Show that the equation $\displaystyle x^3-15x+c=0$ has at most one root in the interval $\displaystyle [-2,2]$
- Show that a polynomial of degree 3 has at most three real roots
- Show that a polynomial of degree n has at most n real roots.

2. Originally Posted by Robb
Dont know how to answer the more general forms of these questions using Rolle's Theorem and mean value theorem, for example;

- Show that the equation $\displaystyle x^3-15x+c=0$ has at most one root in the interval $\displaystyle [-2,2]$
- Show that a polynomial of degree 3 has at most three real roots
- Show that a polynomial of degree n has at most n real roots.

First one:

Suppose it has more than one, for example, 2. Then the mean value theorem says that there is a number in the interior of [-2, 2] where the first derivative is 0. But f'(x) = 3x^2 - 15 = 3 (x^2 - 5), and its roots are clearly outside the interval.

The other two problems are similar. The result you're using here is called Rolle's Theorem, a special case of the Mean Value Theorem.

Good luck!!