Re: Question about roots
If the root z1 is complex and the coefficients of the polynomial are all real, you must somehow get rid of the complex part of the root when it is multiplied by other roots.
for example, if f(x) has complex roots z1, z2, z3, ...:
f(x) = (x-z1)*(x-z2)*....
Just multiplying out the first 2 terms, we get
f(x) = (x^2 - z2 * x - z1 * x + z1 * z2)*....
For the polynomial coefficients to be real, it must be true that
(z2 + z1) is real and z1*z2 is real. This can only be achieved if z2 and z1 are complex conjugates.
It is also easy to see that the complex roots only come in conjugate pairs, so any even degree polynomial can have all of its roots be complex. If there are an odd number of terms, the additional odd root must be real.
Basically the idea is that the only way to obtain a real result when the roots are multiplied together, the roots must be complex conjugate pairs.