In this lesson, students investigate the connection between the factors of a polynomial, its zeros, and the horizontal intercepts of its graph. Students first studied this with quadratics in an earlier grade and now extend that thinking into higher-degree polynomials while making connections between points on graphs and solutions to equations (MP7). Students reason from the zeros of a function to the horizontal intercepts of its graph and vice versa. Students 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 here, students begin to see connections between factors and zeros.
This lesson also offers opportunity for students to use mathematical language about the zeros of a function and the intercepts of graphs. This connection between an equation for a function and the graph of the function is more difficult to see from the standard form, so students work primarily with the factored form in this lesson. In the next lesson, students will use both the factored form and the standard form of a polynomial function to learn more about it, and will translate between the two forms as needed.
- Identify zeros of polynomial functions written in factored form.
- Understand that if factors multiplied together equal zero for a specific value of $x$, at least one of the factors must also equal zero at that value of $x$.
- Let’s investigate polynomials written in factored form.
- I can find the zeros of a function from its factored form.