In this lesson, students investigate polynomials written in factored form and standard form in order to identify information about the graph of the polynomial. They revisit the context of an open-top box made by cutting a square from each corner of a single piece of paper. The volume of this box is most easily expressed by a polynomial written in factored form, and students see the need to find an equivalent expression in standard form when asked to identify the constant term and degree of the polynomial.

An optional activity provides additional practice expanding expressions written in factored form into standard form. Next, students use precise language (MP6) to describe functions and to discuss the similarities and differences between two polynomials whose expressions differ only by a constant multiple. This activity works best when each student has access to devices that can run Desmos, because students will benefit from seeing the relationship in a dynamic way.

Students will use these skills in order to sketch graphs of polynomials expressed in any form, and to understand the end behavior of functions, which will be addressed in following lessons.