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Complex Conjugate Root Theorem

Rule

Complex Conjugate Root Theorem

Let be a polynomial function whose coefficients are real. If a complex number is a root of then the root's complex conjugate, is also a root of

Proof

Let be a polynomial with real coefficients Using the properties of conjugation, the conjugate of the polynomial evaluated at can be rewritten.

Conjugate of the polynomial evaluated at
Conjugate of a sum is the sum of the conjugates
Conjugate of a product is the product of the conjugates
Conjugate of a power is the power of the conjugate
Conjugate of a real number is itself
The conjugate of is
The polynomial evaluated at

It is given that is a root of the polynomial The conjugate of the real number is so the conjugate of the polynomial evaluated at is Comparing this with the equality in the last line of the table shows that the conjugate of is also a root of the polynomial.