Основна теорема алгебри

Introduction

Before we begin, it may be helpful to review complex numbers, real numbers, and how to find linear factors. These are integral skills that will help you to understand this new nifty theorem!

The Fundamental Theorem of Algebra provides the total number of complex roots for any polynomial function.

It states that for a polynomial

$$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$

it has $n$ complex roots, where $n$ is the largest exponent of any variable term in the polynomial. Recall that constant terms don't have a power. Also, we are only looking at polynomials with real coefficients, not complex coefficients.

Let's talk through an example. If we consider the expression, $x-2$, we will see that the degree of this polynomial is $1$. That means that this expression has $1$ complex root.

That was an easy linear polynomial, one with a maximum degree of $1$.

We can take this further, though.

Here is one last practice:

The proof of the fundamental theorem of algebra is quite advanced, so you can click the link.

Now you've learned about the fundamental theorem of algebra; thanks for reading!

The Fundamental Theorem of Algebra

With this new tool in your belt, you will be able to quickly solve for the number of roots in a polynomial function.

In an earlier lesson, we learned about the degree of a polynomial, which is the value of the highest exponent. The Fundamental Theorem of Algebra tells us that the degree is equal to the number of roots of the polynomial. This means that we can easily understand how many roots a polynomial has.

(Video) What does the fundamental theorem of algebra tell us about a polynomial

by Brian McLogan

In this video by Brian McLogan, you will learn about what the Fundamental Theorem of Algebra tells us.

Summary

The Fundamental Theorem of Algebra tells us that the degree of a polynomial will tell us how many zeroes, real and complex, that a polynomial has.

But - remember that all complex zeroes must come in pairs. So, there will always be a positive and negative value for that complex term.