# Complex Conjugates Theorem

How to use the complex conjugates theorem to find the missing zeros: theorem, example, and its solution.

## Theorem

If *a* + *bi* is the zero of *f*(*x*),

then its complex conjugate *a* - *bi*

is also the zero of *f*(*x*).

Then, by the factor theorem,

if [*x* - (*a* + *bi*)] is the factor of *f*(*x*),

then [*x* - (*a* - *bi*)] is also the factor of *f*(*x*).

## Example

The highest degree term of the function is *x*^{3}.

The function is 3rd degree function.

So it has 3 zeros.

The given zeros are 2 and 3 + *i*.

Then 3 - *i* (the conjugate of 3 + *i*)

is also the zero.

So the zeros of the function are 2, 3 + *i*, and 3 - *i*.

So *f*(*x*) = 1⋅(*x* - 2)[*x* - (3 + *i*)][*x* - (3 - *i*)].

Simplify [*x* - (3 + *i*)][*x* - (3 - *i*)].

(3 + *i*) + (3 - *i*) = 6

(3 + *i*)(3 - *i*) = 10

So *f*(*x*) = (*x* - 2)[*x* - (3 + *i*)][*x* - (3 - *i*)]

= (*x* - 2)(*x*^{2} - 10*x* + 6)

Sum and product of the roots of a quadratic equation