共轭根定理

Thus far, we have learned about various theorems and tests to help us determine the roots of a polynomial such as the rational roots test, intermediate value theorem, Descartes' rule of signs, and the fundamental theorem of algebra.

##Introduction

The conjugate root theorem states that if a root of a polynomial is a complex number $a + bi$, then its complex conjugate, $a - bi$, is also a root.

Given the example polynomial: $y = 2x^2 + 5x -21$, after using the quadratic formula, we know its roots are $-\frac{5}{4} \pm \frac{\sqrt{193}}{4}$.

Let's say we were only given the root, $-\frac{5}{4} + \frac{\sqrt{193}}{4}$. Using our new theorem, we know its complex conjugate, $-\frac{5}{4} - \frac{\sqrt{193}}{4}$ is also a root.

Try this example problem for practice!

Conjugate Root Theorem

We've got a nifty theorem that will help with understanding the complex roots of polynomials!

If we know that one of the roots of a polynomial is $a+bi$, then we can assume that the complex conjugate, $a-bi$ is also a factor. Complex roots must appear in conjugate pairs when writing a polynomial with real coefficients.

(Video) Conjugate Root Theorem

by Joel Speranza Math

In this video by Joel Speranza Math, you will learn about what the conjugate root theorem is and how to prove it.

Summary

Introduction

Let $P(z)$ be a polynomial with real coefficients. If $a+bi$ is a root of $P(z)$, so is $a-bi$.

For example, if we have the polynomial, $P(z) = 3x^4 + 5z^3 - 2z^2 + 4z -9$, we can see that all of the coefficients ($3, 5, 2, 4$) are all real numbers.

To find the root of a polynomial, we set it equal to $0$ and solve.

$$0 = 3x^4 + 5z^3 - 2z^2 + 4z -9$$

After doing some math, we will find that the solution to this polynomial is: $z=-0.12 - 1.14i$.

This Theorem allows us to say that in addition to the above root, its conjugate is also a root for this polynomial! Therefore, our other solution is $z=-0.12 + 1.14i$.

These may not be the only solutions, but if given one conjugate as a solution, you can be absolutely sure that the other conjugate is also a solution.

Proof

To understand how this theorem is true, you can watch the video for a walkthrough proof!