Polynomial Practice & Review

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Objective

SWBAT identify zeros and factors of polynomial functions, and use synthetic substitution to evaluate and graph cubic and quartic polynomials.

Big Idea

Polynomial functions have a predictable number of zeros and we can find them with technology, synthetic substitution, or factoring.

Quartic From Homework Check and Fundamentals Review (pun intended!)

20 minutes

Practice Problems

25 minutes

At this point, I will pass out the Polynomial Practice & Review assignment, which has a number of problems taken from our textbook.  In these problems, students will encounter a variety of polynomials having different numbers of real and complex zeros.  In one case there is double root which often gives students trouble (see my comments on multiple roots narrative)

While graphing is not required for the last three problems, I would suggest that students examine a graph in order to gain some insight into any possible integer roots, as well as the number of real vs. complex zeros.  They are free to use a graphing calculator or to quickly plot a number of points by hand using synthetic substitution, depending on what's available and appropriate (MP 5).

Pedagogy:  On days like this, I always make a point of encouraging my students to complete as much of the assignment as they can on their own (MP 1).  Since I allow a fair bit of collaboration on a day-to-day basis, some of the weaker students come to rely too much on their peers.  It's natural, and they don't realize it's happening, but they end up with a somewhat inflated idea of how well they're doing.  I'd rather they struggled today when they are free to ask all the questions they want, than that they find out on the test how much they needed their peers.

The class will be fairly relaxed, students will form small groups, separate, and form others as they find they need help on one problem or another.  I will circulate to answer questions and observe their progress, but I won't give away any of the answers.  The students ought to be able to verify their own work in a variety of ways, and I don't want to encourage them to rely on me for approval.  I'll encourage them to compare their answers with one another, to check them against a graph, to evaluate the function at the suspected zeros, but I won't say, "That's right.  You're done!"  This is one way I help form a habit in my students of carefully attending to precision in their work, as well as encourage perseverance in solving problems since a problem isn't really solved unless you can have confidence that your solution is correct!

Preparing for the Assessment

5 minutes

At the end of class, I like to offer my students some encouragement to continue preparing for the assessment on their own.  Today's lesson should have helped them to identify their strengths and weaknesses, so now they should dedicate some time to addressing those weaknesses.  Please see my Strategy Video, Preparing for an Assessment for more details.

In my view, the textbook is an excellent resource that too many students make too little use of.  I always point out that you can find explanations & example problems in the book, as well as a ton of problem to practice with.  Even better, the answers to the odd problems are in the back, so you can check your work!  So, I tell my students to find the section with the problem that they're struggling with, to read it, study the examples, try some of the exercises problems, and then check their work.

With this particular unit, there are any number of things students might struggle with, but the most likely things are correctly identifying complex roots and making the conceptual connections between equations, numerical solutions, and graphs.  The ability to move flexibly from one representation to another is very important in algebra but it doesn't come easily!