Polynomial Root Theorems
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 as expressed in the following table.
The conjugate of the polynomial evaluated at
The conjugate of a sum is the sum of the conjugates.
The conjugate of a product is the product of the conjugates.
The conjugate of a power is the power of the conjugate.
The 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 Therefore, the conjugate of the polynomial evaluated at is
This equation, along with the last equality in the table, shows that the conjugate of is also a root of the polynomial.
Exercises