Let $p(x)=x^5-4x^4+3x^3-2x^2+5x+1$ and say $a, b, c, d, e$ are the roots of $p$. Find the polynomial whose roots are $abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde$.
By Viete's theorem we just need to find the values of the elementary symmetric functions corresponding to the ten roots. But each such function is a symmetric function of $a, b, c, d, e$, and hence can be written as a polynomial in the $5$ elementary symmetric functions coming from $a, b, c, d, e$, whose values are the coefficients of $p$. Thus it is possible to compute the coefficients of the desired polynomial without explicitly finding the values of $a, b, c, d, e$.
However, this will require $10$ different arduous computations. Is there a nifty way to do this?