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I have known some Sufficient Condition for All the Roots of a Polynomial To Be Real. Is there any sufficient condition that a polynomial of degree $n$ has $n$ distinct roots? For $n=2$, it is trivial.

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Have you read about discriminants? – Dylan Moreland Nov 24 '11 at 2:09
up vote 8 down vote accepted

The condition, over a field of characteristic 0 (or characteristic greater than the degree), is that the gcd of the polynomial and its derivative is 1.

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@Sunni You ask for only a sufficient condition in your post. But note that this is both necessary and sufficient. – Srivatsan Nov 24 '11 at 2:16

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