3.9.3—The result that

A polynomial of degree n with complex coefficients has exactly n complex roots counted with multiplicity. Real roots are included among these; non-real roots occur in conjugate pairs for real-coefficient polynomials.

Use a known root or factor to reduce degree, then use coefficient relationships or conjugacy to find remaining roots. Check multiplicities rather than counting distinct values only.

If a real cubic has root 2 and root 1+i, it must also have 1−i; the three roots account for the degree.

“Three roots” can include a repeated root; distinct-root counting can undercount the degree.