Zeros of a polynomial

[arp]

How do you find the zeros of a polynomial of even degree above 4 whose graph lies above the x-axis?

[EFuzzy]

There's no fool proof method for finding the (exact) complex zeroes, only some tricks for special cases.

[arp]

hm could you give me an example of some of these tricks?

[p_adic]

Examples:

P(x) = x^6 + x^4 + x^2 + 1 is always positive, for real x to find complex zeros note that

P(x) = (x^8 -1)/(x^2 - 1) and hence P(x) is zero when x^8 - 1 = 0 and x^2 - 1 is nonzero.


Q(x) = x^8 + 2x^4 + 1 factors as (x^4 + 1)^2 and so the zeros are the same as those of x^4 + 1 = 0.


R(x) = x^8 + 5x^5 + 5x^3 + 1 = 0 can be solved by dividing through by x^4:

x^4 + 5(x + 1/x) + 1/x^4 = 0

Let u = x + 1/x

Then u^2 = x^2 + 1/x^2 + 2,

(u^2 - 2)^2 = x^4 + 1/x^4 + 2

So x^4 + 1/x^4 = (u^2 -2)^2 -2, hence the equation is

(u^2 -2)^2 - 2 + 5u = 0

which can be solved for u by the quartic formula and then for u.

[goodyfresh741]

Don't go around looking for a solution (by algebraic methods) to the generalized quintic equation. In the 1800's (I'm not sure exactly when) Abel proved that it's impossible to find an algebraic solution to a general equation of degree >=5. In other words, choose any equation of degree >=5, and (with almost 100 % likelihood) it is impossible through a finite number of algebraic operations (addition and subtraction of polynomials from both sides, plus root-taking operations and such) to solve the equation. That can only be done in special cases. If you're interested in reading more about it, there is a "general solution" to the quintic equation using something called Jacobian theta functions (you can also use Weirstrassian theta functions for that purpose) but it's quite high-level math and won't do anything for you on a high school math contest.

[jli]

find all the factors of the coeficient of the one with the largest power

find the factors of the number at the end

divide them in all the possible ways and put a plus/minus in fron of the fractions and those are the possible roots

[goodyfresh741]

[Scrambled]

[ZetaX]

goodyfresh741: can you please give me a link or something like that to a solution for quintic equations using Thetafunctions¿

[bénabar]

[bénabar]

...but Galois was french .

[ZetaX]

I think Abel proofed it for rational polynomials, Galois in a more general case for fields. (But's possible that I'm wrong)

[goodyfresh741]

Yes, Abel proved it first, but Galois proved it for a more general class of polynomials over fields.

[AntonioMainenti]

This guy (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html) proved that the quintic is unsolvable by radicals around 1800 and sent it to Lagrange and others but no one seemed to care or believe him. I know about him because he was a northern Italian whose last name was Ruffini. My father is a northern Italian whose last name is Ruffini We could be related..