Conjugate Zeros Theorem

Videos, worksheets, games and activities to help PreCalculus students learn about the conjugate zeros theorem. What is the Conjugate Zeros Theorem or Conjugate Pair Theorem? How to use the Conjugate Zeros Theorem to factor a polynomial?

What is the Conjugate Zeros Theorem?
The Conjugate Zeros Theorem states that if a complex number a + bi is a zero of a polynomial with real coefficients then the complex conjugate of that number, which is a - bi, is also a zero of the polynomial. It is also called the Conjugate Pair Theorem.

How to use the Conjugate Zeros Theorem to factor a polynomial?
We can use the Conjugate Zeros Theorem to help find the zeros of an expanded polynomial. If we are given an imaginary zero, we can use the conjugate zeros theorem to factor the polynomial and find the other zeros.

How to use the conjugate zeros theorem for polynomials with real coefficients?
Example:
Find a cubic in a factored form with real coefficients, a leading coefficient of 5, and zeros that include 9 and 5i.

How to use the conjugate zero theorem to find all of the complex zeros of a polynomial?
Example:
Find a cubic polynomial in standard form with real coefficients having zeros -4 and 3 + 2i.

The Conjugate Pair Theorem
This video shows that if we know a complex root, we can use that to find another complex root using the conjugate pair theorem.
Example:
Suppose f(x) is a polynomial with real coefficients and zeros:
√3, -i, 5 - 4i, (1 + i)/8
Find three additional zeros of f(x)

Factor Polynomial Given a Complex / Imaginary Root
This video shows how to factor a 3rd degree polynomial completely given one known complex root.
Example:
Find all zeros of:
P(x) = x³ - 4x² + x - 4
given that i is a zero.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.