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].