Fundamental Theorem of Algebra: Explanation and Example

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  • 0:01 Bank Fee Analogy
  • 0:48 Fundamental Theorem of Algebra
  • 2:06 Imaginary Solutions
  • 3:01 Repeated Solutions
  • 3:55 Examples
  • 6:58 Lesson Summary
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Lesson Transcript
Instructor: David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

In this lesson, you will learn what the Fundamental Theorem of Algebra says. You will also learn how to apply this theorem in determining solutions of polynomial functions.

Bank Fee Analogy

This lesson will show you how to interpret the fundamental theorem of algebra. After completing this lesson, you will be able to state the theorem and explain what it means. Before we state the theorem, we will consider the following analogy.

Let's say your bank charges a fee every time you withdraw money from an automatic teller machine. If you withdraw money five times in a particular month, then you will expect five respective bank fees on that month's statement. Let's change this statement by using some mathematical lingo:

If you withdraw money n times in a particular month, then you will expect n respective bank fees on that month's statement.

5 withdrawals = 5 bank fees

The fundamental theorem of algebra is just as straightforward as this banking analogy.

Fundamental Theorem of Algebra

The fundamental theorem of algebra states the following:

A polynomial function f(x) of degree n (where n > 0) has n complex solutions for the equation f(x) = 0.

Please note that the terms 'zeros' and 'roots' are synonymous with solutions as used in the context of this lesson.

That is pretty much it. Now, we should already know that polynomials can be described by their degree. For example, the polynomial x^3 + 3x^2 - 6x - 8 has a degree of 3 because its largest exponent is 3. The degree of a polynomial is important because it tells us the number of solutions of a polynomial.

The theorem does not tell us what the solutions are. It only tells us how many solutions exist for a given polynomial function.

So what good is that? First of all, it is important to understand underlying concepts of any math topics you are learning. In addition, the fundamental theorem of algebra has practical applications. For instance, if you need to find the solutions of a polynomial function, say, of degree 4, you know that you need to keep working until you find 4 solutions.

Imaginary Solutions

It is important to note that the theorem says complex solutions, so some solutions might be imaginary or have an imaginary part. Maybe we should do a quick review of complex numbers.

Complex numbers are in the form of a + bi (a and b are real numbers). The term a is the real part, and the term bi is the imaginary part. If b = 0, then the number is a real number.

Therefore, all real numbers are complex numbers. Let's look at a couple of examples:

In the complex number 2 + 3i, 2 is the real part and 3i is the imaginary part.

In the complex number 25 + 0i, 25 is the real part and 0i is the imaginary part. Because b = 0, the number simplifies to 25.

Repeated Solutions

Before we look at some examples of polynomial functions, let's clarify the concept of repeated solutions. A polynomial function has repeated solutions if it has repeated factors.

A good way to show this is with the function f(x) = x^3. This function has a degree of 3, so based on our theorem, it has 3 solutions. We might see the three solutions better if we show the function in factored form: f(x) = (x)(x)(x). Let's now make the function equal to zero: 0 = (x)(x)(x). If any of the three factors equal zero, then the function equals zero. Therefore, the solutions are x = 0, x = 0, and x = 0. The solution of zero occurs 3 times.

Example #1

Let's start with the polynomial function f(x) = x^2 + 9. In factored form, this function equals (x - 3i)(x + 3i).

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