Originally Posted by

**tonio** Check you understand the following facts:

1) every real polynomial of odd degree has one real root

2) Complex non-real roots of real pol's appear in conjugate pairs, i.e.: if z = x + iy is a root of the real pol. f(x), then also z' = x - iy is a root of f(x) (basic knowledge of basic properties of complex conjugates is needed here)

Thus let f(x) in IR[x] of degree n >= 4 ==> we can write down its roots as follows: r_1, ...,r_n = real roots and z_1, z_1', z_2, z_2',...z_k, z_k' = complex non-real roots.

From here he can write

f(x) = (x-r_1)*...*(x-r_n)*(x-z_1)(x-z_1')*....

Now, check that every pair (x - z_i)(x - z_i') is a real root, and you get what you want.

Tonio