Class will begin with a discussion of how to find the zeros of a cubic function that does not have three real roots. My notes for the details of this discussion are included as a resource. It's important to do this right because it's laying the foundation for the Fundamental Theorem of Algebra and bringing together a number of different concepts.
I'll begin with the equation P(x) = x^3 + 4x^2 -2x - 20 on the board and ask the students how we should go about finding roots. I expect that many students will suggest looking at the graph, so we'll do that using GeoGebra or some other graphing software. The graph will clearly show that the function has only 1 real root at or near x = 2. We'll use synthetic substitution confirm that x = 2 is a zero of the function and then carefully show that there are exactly two more complex roots.
This class example is a good review of yesterday's lesson, but prepares students to move further. First, they will see (if they hadn't already) that the process of synthetic substitution not only confirms that (x - b) is a factor of the polynomial, but yields the coefficients of the other factor, as well. Second, they will have seen a good example of how the acceptance of complex roots allows us to identify exactly n zeros for an n-degree polynomial. This is exactly what we need to begin discussing the Fundamental Theorem of Algebra!
Now that students have seen (another) example of how complex roots/factors can be found, it's time to introduce the Fundamental Theorem of Algebra. By this point, most students should be able to see intuitively that something like the Fundamental Theorem must be true, but it's important to make it explicit for everyone.
In the resources, I've included a page of notes on the line of argument I like to take with my classes. I use Socratic questioning to develop a justification for the theorem based on the characteristics of odd and even polynomials. Along the way, it's important to punctuate the conversation with all sorts of quick examples to show, for instance, why a cubic function couldn't have more than three linear factors.
As a general rule, I avoid drawing any conclusions for my students. If they aren't seeing what follows from the statements we've made so far, I try to draw their attention to different pieces of the puzzle more explicitly and ask more and more pointed questions. Initially, I might ask, "We said before that every odd polynomial must have at least one real root. So, do odd polynomials have exactly n complex roots?"
If the students aren't able to answer, I'll have to be more specific. "If one root is real, is it possible to factor every odd polynomial in some way?" Assuming I get the correct response from the class, I'd then go on to ask, "So, one factor is linear; what must be true about the other factor? Is the other factor an odd or even polynomial? If it's an even polynomial, how many complex roots must it have?" This last question is very pointed and should lead every student to recall the previous step in which we showed that every even polynomial must have exactly n roots. Hopefully, I won't have to be this pointed because I want to leave as much of the thinking to the students as possible!