Students investigate the connection between the factors of a polynomial, its zeros, and the horizontal intercepts of its graph. Students were introduced to these ideas with quadratics in a previous course. In this lesson, they extend that thinking into higher-degree polynomials, using structure to make connections between points on graphs and solutions to equations (MP7). Since these connections are more difficult to see from the standard form, students work primarily with the factored form in this lesson.

Students begin by noticing patterns, given equations of cubic polynomials written in factored form and their graphs. They reason about the zeros of a function and the horizontal intercepts of its graph, and also use the zero product property to infer zeros from factors. Students will not be able to infer a function’s linear factors from its zeros until the Remainder Theorem is proved in a later lesson, but students can begin to see connections between factors and zeros here.

Then students strengthen their understanding by matching equations with graphs or descriptions of graphs, using intercepts and other features such as input-output pairs.

Today’s activity is for students to individually reflect on the norms generated so far. During the Cool-down, students provide feedback on the norms, sharing those they agree with and those they feel need revision or removal. These suggestions will inform the next version of the classroom norms.