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The discriminant of the zero polynomial, P(x) = 0, is zero because it has multiple roots at every point: For all x, P'(x) = P(x) = 0. Thus, the formula for the discriminant gives

Δ = a_0^(0-2) * Π_{all different roots counting multiplicity} (one root - other root)^2

Note that the second term contains an uncountable number of zeroes multiplied together. Dividing out two of them leaves you with an uncountable number of zeroes multiplied together-- i.e. zero.

The above argument can be made more rigorous by adding in all the limits and stuff.

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Q: What is the discriminant of an equation that is zero?
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