Question

Why polynomial of order n must have one root inside the complex?

Answer 1

If zz is a root of a real polynomial, say p(z)=∑nj=0rjzj=0p(z)=∑j=0nrjzj=0, then z¯¯¯z¯ is also a root of pp as p(z¯¯¯)=∑nj=0rjz¯¯¯j=∑nj=0rjzj¯¯¯¯¯=∑nj=0rjzj¯¯¯¯¯¯¯¯¯=p(z)¯¯¯¯¯¯¯¯¯=0p(z¯)=∑j=0nrjz¯j=∑j=0nrjzj¯=∑j=0nrjzj¯=p(z)¯=0. Thus, non-real roots of real polynomials always come in pairs and their number is thus even.

There is no restriction (but the degree) on the number of real roots, though; it is possible that the polynomial of degree 44has 33 real roots too, like x2(x−1)(x−2)x2(x−1)(x−2).