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.