No. of real roots of a polynomial?

Forum for the GRE subject test in mathematics.
hadimotamedi
Posts: 34
Joined: Thu Dec 30, 2010 4:36 am

No. of real roots of a polynomial?

Postby hadimotamedi » Thu Dec 30, 2010 4:39 am

Dear All
Can you please let me know if there is a simple method for obtaining the number of real roots of a polynomial?
Thank you

enork
Posts: 33
Joined: Fri Sep 18, 2009 3:16 am

Re: No. of real roots of a polynomial?

Postby enork » Fri Dec 31, 2010 3:32 am

If you're asking about a general polynomial with complex coefficients I don't know of an easy way to tell. If you're asking about a polynomial with real coefficients, the easiest way I know of is to consider the polynomial as a function over the reals, then find the critical points and evaluate at those points to see how many times the function changes sign.

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

Re: No. of real roots of a polynomial?

Postby owlpride » Fri Dec 31, 2010 5:46 pm

The critical points thing is a neat idea but it requires knowledge of the roots of the derivative, which may be just as hard to find as the roots of the original polynomial.

I am pretty confident that there is no easy way in general to determine the number of real roots of a polynomial.

freeabelian
Posts: 9
Joined: Sat Jan 01, 2011 5:35 pm

Re: No. of real roots of a polynomial?

Postby freeabelian » Sat Jan 01, 2011 5:38 pm

This is not a trivial problem.

There are ways you can guarantee it has some number of roots, for instance, if the degree of the polynomial is odd, it must have a real root (as the only ireducible polynomials over \mathbb{R} the squares).

Potentially the best way isn't to look for critical points, as that doesn't really givey ou any information at all (for example, the parabola y=x^2+1 has a critical point, but no zeros) but to use the mean value theorem. Find a value where it is positive, find a value where it is negative, and that's that, there is a zero in between.

You can also use limit behavior, for example, if the function is positive at x=0, and goes to -\infty as x \rightarrow \pm \infty, you must have a zero as well.

Lastly, the rational root theorem can help you look for roots if you need to.

Lastly, again, a general polynomial with complex roots will actually yield TWO polynomials if you're looking for real roots, as you have to cancel out the imaginary and the real part of each component. This might be helpful if you needed to find real roots of a polynomial in C[x].




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