3.4.1 Imaginary Roots

. . Imaginary Roots For a quadratic equation with real coefficients, if α β + i is a root, then α β − i is also a root. In this section we shall prove that this is true for higher degree polynomials as well. We now prove one of the very important theorems in the theory of equations. Theorem . (Complex Conjugate Root Theorem) If a complex number z is a root of a polynomial equation with real coefficients, then its complex conjugate z is also a root. Proof Let P x a x a x o ( ) =  be a polynomial equation with real coefficients. Let z be a root of this polynomial equation. So, P ( z ) = . Now P z