Calculate root interval of $x^5+x^4+x^3+x^2+1$ So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$
I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.

The common factoring formula $x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \cdots + x^2+1)$ tells us that your $f(x)$ is equivalent to $g(x)=\dfrac {x^6-1}{x-1}$ as long as $x \not = 1$. Setting $g(x)=0$ yields $x^6-1=0$. Clearly the only real solutions to this are $\pm1$. But only $-1$ is a zero of $g$ and of $f$, so $f$ has only one real zero.
