Conjugate Root Theorem: Definition & Example

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The conjugate root theorem is a powerful theorem in mathematics. We will explore this theorem and its parts. We will also look at many examples of applying this theorem in different situations.

Complex Conjugates

Let's talk about the conjugate root theorem in mathematics! This is a very interesting and important theorem in mathematics that aids in finding the complex roots of a polynomial. Before we get to the actual theorem, we need to familiarize ourselves with the numbers that are involved with this theorem.

To do this, let's start by looking at the first word in the theorem: conjugate. As an adjective, ''conjugate'' is to be joined together, normally in pairs. Therefore, it should come as no surprise that when we talk about complex conjugates in mathematics, we are talking about a pair of numbers! Complex conjugates are a major part of the conjugate root theorem, so we definitely want to be familiar with them.

Complex conjugates are two complex numbers, so they have the form a + bi, where a and b are real numbers and i = √ -1. We call a the real part of a complex number and bi the imaginary part.


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Two complex numbers are conjugates of each other if they have the same real part and their imaginary parts are negatives of each other. That is, the complex conjugate of a + bi is a - bi.


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For example, the complex conjugate of 5 + 3i is 5 - 3i, and the complex conjugate of 1 - 4i is 1 + 4i. Finding the complex conjugate of a complex number is simple; you just change the sign of the imaginary part of the number.

Okay, now that we are familiar with the types of numbers that the theorem involves, let's dive into the theorem itself!

Conjugate Root Theorem

The conjugate root theorem states that if the complex number a + bi is a root of a polynomial P(x) in one variable with real coefficients, then the complex conjugate a - bi is also a root of that polynomial.


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Fancy, but you may be wondering how this is useful. Well, suppose you are trying to find the roots of a polynomial, and, as you are solving, you find out that the complex number a + bi is a root of the polynomial. Well, as soon as you find this out, you automatically know another root! By the conjugate root theorem, you know that since a + bi is a root, it must be the case that a - bi is also a root.

For example, if 1 - 2i is a root, then its complex conjugate 1 + 2i is also a root. This theorem just saved you the time and hassle of having to find that root in more complicated ways. As you can see, this theorem is very useful when it comes to solving polynomials.

Examples

Another instance when this theorem comes in handy is in analyzing a polynomial. For instance, suppose you are working on a project, and you come across a polynomial with highest exponent two. This tells you that the polynomial has two roots. You find one root to be a real number with no imaginary part. Is it possible that the other root could have the form a + bi, where b is not zero?

Hmmm, let's think about this. The polynomial only has two roots, and one of them has been found to be real with no imaginary part, so the polynomial only has one other root. If that other root were a complex number a + bi, where b is not zero, then by the conjugate root theorem, its complex conjugate must also be a root, but wait! That would result in three roots for the polynomial, and we know the polynomial only has two roots. Therefore, we can logically conclude that the remaining root of the polynomial must be real. Pretty neat, huh? We were able to determine the type of root of the polynomial without actually finding it!

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