Wikipedia:Reference desk/Archives/Mathematics/2010 December 9

= December 9 =

Sum of Roots of Polynomial
Hello. How can I prove that the sum of all roots (real and complex) of a polynomial is $$-\frac{b}{a}$$ where a is the leading coefficient and b is the coefficient of the variable of second highest degree? Thanks in advance. --Mayfare (talk) 01:19, 9 December 2010 (UTC)
 * It's not true, consider x3-1. It is true if you count all roots, including complex, which can be seen by factoring the polynomial.--RDBury (talk) 02:29, 9 December 2010 (UTC)

You're correct. Please pardon me. I've rephrased my question. --Mayfare (talk) 05:17, 9 December 2010 (UTC)
 * For linear problems, this is self-evident. For quadratics, it can be deduced via the Quadratic formula.  I don't know how to derive it for higher degrees, but mathematical induction might be a helpful technique.  -- N  Y  Kevin  @326, i.e. 06:49, 9 December 2010 (UTC)

Consider a general polynomial
 * $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = a_n (x-\alpha_1)(x-\alpha_2) \dots (x - \alpha_n), \,\!$$

where the zeros αi are members of C. The coefficient of xn &minus; 1 on the LHS is an &minus; 1. On the RHS, each term involving xn &minus; 1 is obtained by selecting x (n &minus; 1) times from the n brackets and one of the −αi. Hence, comparing coefficients of like terms on both sides,
 * $$-a_n (\alpha_1 + \alpha_2 + \dots + \alpha_n) = a_{n-1}. \,\!$$

The result follows with algebraic manipulation. — Anonymous Dissident  Talk 07:04, 9 December 2010 (UTC)

For more information on this subject, see elementary symmetric polynomial and Viete's formulas. Eric. 82.139.80.124 (talk) 08:29, 9 December 2010 (UTC)

Name this Surface
I've got a surface given by the equation x2 + 2y3 – 3xyz = 0. The z ≠ 0 constant sections give loops with a node on the z-axis. The z = 0 section gives an ordinary cusp (a.k.a. a semi-cubical parabola), with the cusp point at the origin. I've drawn the pictures, and it's (diffeomorphic to) a very recognisable surface; I just don't remember its name. Any suggestions? — Fly by Night  ( talk )  13:54, 9 December 2010 (UTC)
 * Probably obvious but MathWorld lists some surfaces, and you might try Encyclopédie des Formes Mathématiques Remarquables since it would have pretty much any named surface.--RDBury (talk) 20:41, 9 December 2010 (UTC)
 * I tried MathWorld, and couldn't find it. The French reference isn't all that helpful because, as my original post says: "I just don't remember its name", and there doesn't seem to be an option to search for the formula. I did find a surface that look just like it, called the  trichter surface  on the German website Algebraic Surface Gallery ; but its equation if x2 + z3 – y2z2, which is order four while mine is order three. — Fly by Night  ( talk )  23:22, 9 December 2010 (UTC)
 * I've seen the surface listed as S23 somewhere, possibly a reference to Cayley's classification of Cubic surfaces .--Salix (talk): 20:11, 10 December 2010 (UTC)
 * Found it University of Turin Models--Salix (talk): 20:32, 10 December 2010 (UTC)
 * That's exactly it! I can't thank you enough... — Fly by Night  ( talk )  16:24, 11 December 2010 (UTC)