Today we are going to start talking about roots of polynomial functions. This lesson will take a few days because we will be working with functions that have real, imaginary and repeated roots. Our focus today will be functions with real roots. In finding these roots we will be using graphing software, factoring, and division (both long and synthetic). My students have used synthetic division before and are usually very familiar with the algorithm, but they usually need a reminder that it "works" because it is merely the same algorithm as long division, but with the focus solely on the coefficients.
For the launch, show the second slide of the PowerPoint (Notes - Roots of Polynomial Functions.pptx). This will get students thinking about the concepts that they will be working with during the lesson. I would have students go over these definitions with their group and then you can discuss as a class.
Then show students the third slide and have them brainstorm ways to find the zeros of the given polynomial. Again, have them brainstorm with some classmates before sharing with the class. I would imagine that a precalculus student would be able to think of factoring and graphing as possible ways of finding zeros.
I usually find that graphing is the most common approach to finding the zeros of polynomial functions, so have students start with that method for slide 4 of the PowerPoint (Slides 4 to 7). Let them use graphing software to find the zeros. Two of the roots are whole numbers, but the third one would be given on most graphing calculators as 0.47826. Now challenge students to see if they can find the exact value of this last root. Ask them if the graph is sufficient to find the exact value.
Now the students must use a different method. A student may suggest factoring since it probably came up during the Launch. While a great method, it would be tough to factor 23x^3 – 57x^2 - 47x + 33. Give them a few minutes to come up with a way to get around this. How can they factor without using guess and check?
If students are stuck, you might want to use an analogy with numbers (one of my favorite problem solving strategies - Simplify the Problem). Suppose you know that 101 and 17 are factors of 44,642. How could you find the other factors? Once students understand that division can be used, go back to the polynomial problem and press them to see if they can use their whole number zeros from their graph to find factors of the polynomial.
At this point you can divide the original polynomial by (x-3) and then factor the remaining quadratic function. Of course, you can also divide by the other factor (x+1). Then students will be left with the last factor and will be able to find the exact value of the last zero. After using long division, many students usually remember the synthetic division algorithm. If no one brings it up, you may want to remind them of it.
Slide 6 of the PowerPoint (Slides 4 to 7) instructs students to use long division to divide two numbers (where there is a remainder). Slide 7 asks students to divide two polynomials (there is also a remainder). This way you can use the analogy between the two processes and discuss what to do when there is a remainder for polynomials (i.e. the last term of the quotient will be the remainder divided by the divisor).
On the last slide of the PowerPoint (Slide 8), there is a final question that is a good summary of the work that we have done today. Although it is procedurally known by students, the relationship between factors and zeros is probably not conceptually obvious. I think this is a good way to summarize the entire lesson since we were transitioning from factors to zeros (or vice verse) several times throughout the lesson. While presenting the last slide, give students 5 minutes to write down a thoughtful response to the question. If they are stuck, you could have them look at a special case and see if it can get them anywhere. After the five minutes, have three or four students share their responses. After each response, have another student in the class summarize their explanation to see if they truly understand the concept.