amparvardi wrote: The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$. Prove that $-2 <p + q + r < 0$. Let $a,b,c$ be roots so we have $0<a,b,c<2$ or equivalently $-1<1-a,1-b,1-c<1$ so we have $|(1-a)(1-b)(1-c)|<1$ which means $|p+q+r+1|<1$ and hence $-2<p+q+r<0$

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