The polynomial intersects the x axis at the polynomial's zeroes.
If a and b are the real roots of the polynomial then p(x)=(x-a)(x-b)f(x), where f(x) has complex zeroes only.
Also ((x-a)^m)((x-b)^n)f(x) has only two zeroes (m and n are integers) when f(x) only has complex zeroes or f(x) also has at least one zero matching a or b, and its remaining zeroes are complex.