This unit extends students’ previous work with linear and quadratic functions as they investigate polynomials of higher degree. Students rewrite polynomials in different forms, recognizing the benefits of the various forms for their ability to reveal the structure of key features of their graphs.

The unit begins with an introduction to two situations that can be modeled by a polynomial function. Students build their understanding of what polynomials are and what their graphs can look like. Certain aspects, such as end behavior, will be important in a later unit when students explore the end behavior of rational functions.

Graph of polynomial function y = x squared, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes down through (-50 comma 2,500), descending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (50 comma 2,500).

Graph of polynomial function y = x cubed, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes up through (-20 comma 8,000), ascending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (20 comma 8,000).

Graph of polynomial function, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through (-10 comma 10,000), descending in a smooth curve through the origin, up into Quadrant 1 and ascends in a smooth curve to the upper right through (10 comma 10,000).

Focusing on functions expressed in factored form and their graphs, students connect that a factor of means is a zero of the function and is a horizontal intercept. The effect of the degree and leading coefficient on end behavior is established along with the effect of multiplicity on the shape of the graph near zeros of the function. Taking in all of these features, students learn to sketch polynomial functions expressed as a product of linear factors.

In a previous course, students used the distributive property to multiply factors, and also factored quadratics. Opportunities to review these skills and apply them to polynomials of higher degree are embedded throughout the unit and in practice problems. This prepares students for the final section where students divide a polynomial written in standard form by a suspected factor. From there, the connection between division and multiplication equations is used to establish the Remainder Theorem. This allows the conclusion that if a polynomial has a zero at , then it must also have as a factor.