Complex Conjugate Root Theorem - Examples and Consequences

Examples and Consequences

• The polynomial x2 + 1 = 0 has roots ±i.
• Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue.
• The polynomial

has roots

and thus can be factored as

In computing the product of the last two factors, the imaginary parts cancel, and we get

The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.