Question

Does a polynomial of degree n necessarily have n real solutions? Does it necessarily have n unique solutions, real or imaginary?

Answer 1

No. A polynomial equation in one variable of degree `n` has exactly `n` Complex roots, some of which may be Real, but some may be repeated roots.

Explanation:

For example, `0 = x^4+2x^2+1 = (x-i)^2(x+i)^2` has roots `i`, `i`, `-i`, `-i`.

If the polynomial has Real coefficients, then any Complex roots occur as conjugate pairs. So if a polynomial of degree `n` has Real coefficients, then it has `n - 2k` Real roots and `2k` non-Real Complex roots for some integer `k >= 0` counting repeated roots.