PR4 review

[AbbyANoble]

Tagging @siwelwerd and @kathypinzon since y'all did some review of this section previously.

In looking at PR4 to write the Check It and My Open Math homework, I have some concerns about the section. Some thoughts at first read:

1. Definition 4.4.3 feels more like an observation than a definition. Real zeros are the real values c such that f(c)=0. They correspond graphically to x-intercepts.
2. After Theorem 4.4.4: Factor Theorem, I think there needs to be more motivation/clarity about how this (along with FTA) means we can write any polynomial as a product of linear factors. Then have some practice where all the roots are real, practice factoring.
3. Activity 4.4.6 assumes they know to connect the polynomial division from the last section to what they are doing here. Maybe "Use polynomial division to find..." instead of "this fact". Also saying "it turns out" feels vague and handwavy. How do we know that? Maybe "Given that x-3 is a factor of f(x)". Then in (b), I'd prompt to continue factoring, asking to rewrite f(x) as a product of only linear factors. Then add a question asking them to actually state the zeros and not just say how many there are.
4. In Activity 4.4.8 to Activity 4.4.10, the focus is on degree and number of zeros, but doesn't really highlight actually finding the zeros. (And all the MC answers are the same.)
5. Definition 4.4.11 mentions the factored form of the polynomial, but we haven't really talked about it yet. (Hence my suggestion in number 2.)
6. I think Observation 4.4.12 would be better as an activity to practice the idea than observation where students are told the answer.
7. Activity 4.4.13 feels out of the blue. Maybe ask to write as a product of linear factors, with a followup about what those factors look like? How this "doesn't factor". To motivate that not all zeros are real zeros. Also, this might make more sense with the other complex stuff near the end of the section. It's basically the same as Activity 4.4.17.
8. An activity where a polynomial with some complex roots are given. Then ask to find degree. Then look at a graph and realize there are fewer x-intercepts than the degree. Maybe multiplicity is the issue? Factor and that doesn't lead to repeated real roots. Conjecture there must be something else going on, enter the complex root idea. Point out the theorems earlier didn't specify real roots, but that was what we had been looking at.
9. Add in questions at the end that bring everything together more. Like the check it parts 1 and 2 -- https://teambasedinquirylearning.github.io/precalculus/2024/exercises/#/bank/PR4/1/. Except task 1 is always the same situation, so I think that needs some varying.

[AbbyANoble]

Also wondering what kinds of polynomials we want students to be able to determine the zeros of. If we aren't doing the rational root theorem, then we need the polynomials to be somewhat easily factored and/or of smallish degree. And with the inclusion of PR3 (polynomial division), it looks like we are wanting them to factor "bigger" ones. But how are they getting the first zero? Are we telling it to them? Are we leaning on graphical methods to spot some real zeros first?

And then in the current check it (https://teambasedinquirylearning.github.io/precalculus/2024/exercises/#/bank/PR4/15/), task 1 is always a quartic with two real and two imaginary roots. Some of them have zero as a root, so the factoring isn't bad. But I'm still not sure how we are motivating students to get started in other cases.

Going ahead and tagging @tdegeorge too now since I'm asking about the objective of the section!