How to Use the Fundamental Theorem in Algebra

The Fundamental Theorem of Algebra is often taught after the basics of algebra have been established, and is essential to solving polynomial equations with complex coefficients. This theorem, proven by Gauss, takes some of the guesswork out of knowing whether you've found the correct answer.

Instructions

1. Understand the definition. The Fundamental Theorem of Algebra states that any polynomial of degree n over a algebraically complete field has n not necessarily distinct roots.

2. Determine applicability. If you're not working with complex numbers, the fundamental theorem won't help you.

3. Factor your complex numbers down to the point where they no longer can no longer be factored down.

4. Check against the Fundamental Theorem, wherein the highest degree dictates the number of factors within that number. If you have more, or less, than that number, you've messed up. Start over.

Tips & Warnings

• Factoring isn't all that hard, but a refresher of the process might be in order: Combine like terms and evaluate the remaining terms for common elements (or factors). Simplify by removing these factors until you can't simplify any further.

• Complex numbers are numbers that have the form "kx^n+i" where k and i are constants, x is a variable and n can be any value. The "i" can be equal to zero, and so not be represented in the equation.

Resources

• Examples of the Fundamental theorem, from SOS math
• Factoring resource at Thinkquest