Complex Conjugates Theorem

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).

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

Find the function that satisfy the given conditions. Highest degree term: x^3, Known zeros: 2, 3 + i

The highest degree term of the function is x3.

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)(x2 - 10x + 6)

Sum and product of the roots of a quadratic equation

Multiplying complex numbers

Multiplying polynomials