## Problem 1

Two roots of the real quartic $x^{4}-18 x^{3}+a x^{2}+200 x-1984=0$ have product -32 . Find a.


## Solution

Let the two roots satisfy the quadratic $\mathrm{x}^{2}+\mathrm{hx}-32=0$ (we have not yet shown that $\mathrm{h}$ is real). The other two roots have product $-1984 /-32=62$. Let them satisfy the quadratic $x^{2}+k x+62$. So $x^{4}-18 x^{3}+a x^{2}+200 x-$ $1984=\left(\mathrm{x}^{2}+\mathrm{hx}-32\right)\left(\mathrm{x}^{2}+\mathrm{kx}+62\right)=\mathrm{x}^{4}+(\mathrm{h}+\mathrm{k}) \mathrm{x}^{3}+(\mathrm{hk}+30) \mathrm{x}^{2}+(62 \mathrm{~h}-32 \mathrm{k}) \mathrm{x}-1984=0$. Equating coefficients: $h+k=-18,62 h-32 k=200$. Solving, $h=-4, k=-14$. Hence $a=86$.

