Finding integer part of a polynomial root If a polynomial $f(x)=x^3+2x^2+x+5$ has only one real root $\omega$ then integer part of $\omega = -3$.
It appeared in my school exam and I couldn't solve it completely but somehow tried to find the range in which the root lies. But it was tough to show that $[\omega] = -3$.

$f'(x)=3x^2+4x+1$
The derivative vanishes for $x=-1,-\frac{1}{3}$ It is positive for $x\in(-\infty,-1)$
$f(-3)=-7$
$f(-2)=13$
Thus, by the intermediate value theorem there must be a root in $(-3,2)$ such that $f(\omega)=0$. Thus, it has an integer part of $-3$
