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 $[\omega] = -3$.

$$f(\infty)\to\infty,f(-\infty)\to-\infty,f(0)=5$$ Roughly sketch them:

Red one is possible since blue one has 3 roots. So it is now known that the root is negative.
Now sum of roots is $-2$, product of roots is $-5$, judging we get the roots is nearby $-1$ to $-10$ atmax. Try values at those x and see where it changes from positive to negative and between them form the graph there will be a root.
