In this lesson, our instructor Vincent Selhorst-Jones teaches the Fundamental Theorem of Algebra and the n Roots Theorem. Youll learn about hidden roots and where the imaginary roots are located. Vincent also goes through a proof sketch of the n roots theorem and provides various comments on multiplicity, complex numbers, complex coefficients, and the existence theorem.
This lesson will not be like a standard lesson: there will be hardly any numbers, and no examples at all. This is because we're learning some interesting ideas from advanced math.
The Fundamental Theorem of Algebra says, "Every polynomial of degree n > 0 has at least one root in the complex numbers."
Notice that "in the complex numbers" does not mean the root can't be real. An entirely real number is contained in the complex numbers because they contain both real and imaginary numbers.
The Fundamental Theorem of Algebra allows us to create the n roots theorem: "A polynomial of degree n has exactly n roots in the complex numbers (these roots are not necessarily distinct)." Equivalently, we can also write this as: "For any polynomial
P(x) of degree n > 0, there exist n complex numbers z1, z2,..., zn (not necessarily distinct) and a constant number a such that
P(x) = a (x − z1)(x−z2) …(x−zn).
Some important things to note about the above theorem:
While a polynomial of degree n has n roots, they are not necessarily all distinct. We can have roots that "repeat" multiple times. This idea is called multiplicity.
The above theorem is only true if we're allowing complex numbers. If we only stick to the real numbers, there are many polynomials that cannot be broken up into linear factors.
While it was not explicitly stated, the theorem holds whether the coefficients of the polynomial are real or complex numbers.
The Fundamental Theorem of Algebra and the n roots theorem are existence theorems. However, they do not tell us how to find these roots. All they do is tell us that they're out there somewhere. They guarantee existence, but that's all.
Fundamental Theorem of Algebra
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.