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Given a polynomial of degree four: $ax^4+bx^3+cx^2+dx+e$, with $a,b,c,d,e$ real and $a\neq 0$, how do I derive the condition for there to be exactly distinct 3 real roots (i.e., one root is repeated)? I know that the discriminant is zero when there is a double root. But how do I enforce the condition that there be only one double root?

If this is known, a link to the resource would be appreciated. If not, helpful guidance in how to proceed will be nice. If you already know the answer, that would be great!

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If $\alpha$ is a root of order $m>1$, then $(x-\alpha)^{m-1}$ must be a common factor of your polynomial $p(x)$ and its derivative p'(x). So you run Euclid's algorithm to find d(x)=gcd(p(x),p'(x)). It must be linear in order to $p(x)$ have exactly one (necessarily) double root. Check that the roots of $p(x)/d(x)^2$ are real to make sure that there are 3 real roots.

I don't know, if this is the kind of answer that you wanted, though?

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    Yes. A general quartic will not have a double root, so a constant gcd is to be expected. You get a condition for the the existence of a double root by setting that quantity to zero. If the gcd were of degree >1 (you don't want that to happen), then the polynomial of the previous step would also be zero.2011-07-10