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!