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Show that the Discriminant of the product of two polynomials in $\mathbb{C} [x]$ is the product of the discriminants multiplied by the square of their resultant, Use the Usual definitions from Discriminant,

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    @GerryMyerson No problem.2013-06-12

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[comment promoted to answer, at suggestion of @Julian Kuelshammer]

This comes from rolling out the formulas for discriminant and resultant. Discriminant of $f$ involves a product of differences of roots of $f$, resultant of $f$ and $g$ involves a product of differences of roots of $f$ and roots of $g$, and the roots of a product are the union, with multiplicities, of the roots of the individual polynomials.