Next, I give students time to work individually on problem 1 of Higher Degree Polynomials 2. It's important to provide some individual time for students to complete at least 1a and 1b on their own. Many will have incorrect solutions, but that's okay for now (see below).

I expect a number of students to fail to identify the third root in the first function. When students fall into this trap, I don't give away that they've done something wrong, but innocently suggest that they might find it helpful to plot the points from the table on the given graph. This should force them to recognize that the graph must cross the x-axis more than twice.

Another potential misconception is to think that since the function has three real roots, it must be a cubic function. In this case, I might point to student work from problem 2a of the previous lesson (a quintic polynomial with three real roots). This will be enough for many students to recognize their mistake. Alternatively, I might look for two confident students with conflicting answers and point out the difference. Typically, this will start a debate that will draw in enough other students for the truth to win out (MP 3). If not, I'll step in.

By the time all the students have an answer written for 1a and 1b, I will allow them to begin comparing answers and collaborating on the rest of the handout.

After working in groups until about 5 minutes are left in this section, I'll call for a quick summary discussion. Student volunteers will share out their answers to the various parts of problem 1. Hopefully, there will be universal agreement, but if not, I'll help students to see the reason behind the correct answer.