Complex Conjugate Theorem

Definition

The Complex Conjugate Theorem states that if a polynomial has real coefficients, then any non-real complex zeros must occur in conjugate pairs. If $a + bi$ is a zero, then $a - bi$ must also be a zero.

5 Must Know Facts For Your Next Test

1. A polynomial with real coefficients cannot have an odd number of non-real roots.
2. If $2 + 3i$ is a root of the polynomial, then $2 - 3i$ is also a root.
3. Complex conjugates have the same real part but opposite imaginary parts.
4. The product of complex conjugates is always a real number.
5. To find all zeros of a polynomial with real coefficients, once one complex root is found, its conjugate must also be included.

Related terms

Polynomial:

An expression consisting of variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents.

Zero of a Polynomial: A value for which the polynomial evaluates to zero. Also known as 'root' or 'solution'.

Complex Number: A number in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$.