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7
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polynomials with the same discriminant
If you write your cubic as $p(x) = x^3 + rx^2+sx+t$, then the discriminant is $\Delta=r^2s^2-4s^3-4r^3t-27t^2+18rst$, which is quadratic in $t$. So the fact that $p(x+a) + b$ has the same discrimi …