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.

Let $f(x)=x^5+x^4+x^3+x^2+1$.
Hence, $f'(x)=x(5x^3+4x^2+3x+2)$ and since $(5x^3+4x^2+3x+2)'=15x^2+8x+3>0$,
we see that the polynomial $5x^3+4x^2+3x+2$ has one real root $x_1$ and this root is negative.
Thus, $x_{min}=0$ and $x_{max}=x_1$. 
But $f(0)>0$, which say that $f$ has an unique real root and since
$$f(-1.25)f(-1.24)<0,$$
we get that this root placed in $(-1.25,-1.24)$.
