a post from Erich Friedman <efriedma@stetson.edu> --rich greetings funsters,
consider the problem of which monic 4th degree polynomials x^4 + a x^3 + b x^2 + c x + d have the property that some line is tangent to the graph at 2 different places. there are some hard ways to think about this, equating slopes with average slopes, but the solution is obvious once you realize that these are the polynomials with 2 inflection points, so take 2 derivatives and look for 2 real roots. turns out the condition is 8b < 3a^2. the fact that c and d don't matter should be clear because of horizontal and vertical shifts. --I did not understand that. My way of solving it (which maybe Erich Friedman disagrees with and is wrong, but I'll say it anyhow?): Let the quartic polynomial be Q(x). Let the line be Ax+B. Then Q(x)-Ax-B is another monic quartic, which is tangent to the x-axis at 2 places. It would seem to me, these are precisely the quartics ((x-J)*(x-K))^2. And that is the solution of the problem: there must exist A and B such that Q(x)-Ax-B has this squared-quadratic form.
all that is background for my real question, one i am having trouble answering. which monic 6th degree polynomials x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F have a line that is tangent to the graph at 3 different places?
if such a line y = U x + V exists, then the polynomial x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F - Ux - V is tangent to the x-axis at 3 points, which means it is of the form (x - r1)^2 * (x - r2)^2 * (x - r3)^2 for some distinct r1, r2, r3, but which A, B, C, and D accomplish that? i can't seem to solve the resulting equations. is there an easier way of doing this? erich friedman --similarly, let the sextic be S(x) and let the line be Ax+B. The solution to the problem is to demand there exist A and B such that S(x)-Ax-B adopts the squared-cubic form ((x-J)*(x-K)*(x-L))^2. Now I don't know whether you consider my solutions "solutions" but let me say this. I have given a parameterized description (parameters being J,K,L,A,B) of every monic sextic, and it yields only the monic sextics, that do this. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)