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.

EDITED for correct function.
$f(x) = 1+x^2+x^3+x^4+x^5$
is positive for x>0.
So consider $x < 0$:
One can write $f(x) = (1+x^2)(1+x^3) + x^4$ which is positive at least for $(1+x^3) >0$, i.e. $x > -1$.
$f'(x) = 2x+3x^2+4x^3+5x^4 = x(2+3x)+x^3(4+5x)$
so this is positive at least for $x < -4/5 $ since then all terms are positive.
This means that $f(x) = 0$ can only happen at $x < -1$  and since $f(x)$ is rising there, we have that there can only be one real root, which is  at  $x < -1$.
Now this root can be further locked in.  Observing e.g. that $f(-1.5) = -85/32 <0$, we can give an interval for the root at (-1.5 , -1). This can obviously be improved, knowing that there is only one real root and that f(x) is monotonously increasing  in this interval.
