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

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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. --NYKevin @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 α_i are members of C. The coefficient of x^{n − 1} on the LHS is a_{n − 1}. On the RHS, each term involving x^{n − 1} is obtained by selecting x (n − 1) times from the n brackets and one of the −α_i. 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 [1].--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)