# If the sum of two roots of x:4+2x:3−8x:2−18x−9=0 is 0, find the roots of the equation

If the sum of two roots of
${x}^{4}+2{x}^{3}-8{x}^{2}-18x-9=0$
is 0, find the roots of the equation
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Payton Mcbride
Generally, if there's two roots whose sum is zero, then it means that two factors are x−a and x+a, which means that ${x}^{2}-{a}^{2}$ must be a factor. So clearly
$\left({x}^{2}-{a}^{2}\right)\left({x}^{2}+bx+c\right)={x}^{4}+2{x}^{3}-8{x}^{2}-18x-9=0$
Find the values of ${a}^{2}$, b, and c that satisfy the left equality, and you'll have found factors that you can then solve for all roots. (This works even if there's no rational root)