This section introduces students to polynomials in different forms and connects features of the equations of polynomials to their graphs.

Students begin by exploring two contexts that can be modeled by polynomials. First students cut out identical squares from each corner of a rectangular sheet of paper and then fold it to create an open-top box. The volume of the box is a function of the side length of the square cutouts and is most easily expressed by a cubic function written in factored form.

Next students consider an investment that earns interest each year and is most easily modeled by a polynomial written in standard form. These two examples showcase how the form of a polynomial can make certain features more clear.

Students continue to explore polynomials written in standard form by first matching equations to graphs and then using technology to explore how to write equations that have specific features in their graphs.

Finally, students consider similarities between arithmetic operations with integers and arithmetic operations with polynomials. They generalize that, as with integers, addition, subtraction, and multiplication of two polynomials results in another polynomial. This is not always the case when dividing two polynomials.

In this section, students see how the standard and factored forms of a polynomial each highlight different features, and they make use of structure to move flexibly between them. Students begin by noticing patterns between the factors of a polynomial and the -intercepts of the graph of the polynomial. They extend their previous work with quadratics to generalize that if factors multiplied together equal 0 for a specific value of , then at least one of the factors must also equal 0 at that value of .

Graph of a polynomial function, xy-plane.  Horizontal axis, scale -6 to 5 by 1’s. Vertical axis, scale -30 to 20, by 10’s. Function starts as increasing in Quadrant 3 and ends as increasing in Quadrant 1. The function has roots that are increasing through (-5 comma 0), decreasing through (-1 comma 0) and increasing through (3 comma 0). The function has a local maximum at (0 comma 8), local minimum at (3 comma -4) and intercepts the y-axis at (0 comma -15).

Next students see that while factored form is useful for finding zeros and -intercepts, standard form is more useful for determining the degree and constant term of a polynomial. They build on their knowledge of the distributive property to rewrite polynomials from factored to standard form.

Finally, students consider the effect of a constant factor on the graph of the polynomial. They write possible equations for polynomials with specified horizontal intercepts, and consider appropriate viewing windows when using technology to graph polynomials.

In this section students are introduced to polynomial division. They begin by using diagrams to help reason about the distributive property. Next they explore parallels between long division with integers and long division with polynomials. They compare using the algorithm for long division with the reasoning used in completing a diagram.

Next, using the Information Gap structure, students put together everything they have learned about polynomials to sketch a graph of a polynomial written in standard form, given one linear factor. They must first use division to rewrite the polynomial as the product of linear factors, and then take into account the degree, zeros, and multiplicity of each factor in order to sketch a graph of the polynomial, including end behavior.

Students continue making connections as they compare the meaning of the remainder when dividing integers and the meaning of the remainder when dividing polynomials. They use the fact that if is a factor of a polynomial , then to determine that if then is not a factor of . Students are also briefly introduced to the Remainder Theorem that states: When is divided by , then the remainder is equal to .

In this section, students use characteristics of polynomial functions such as factors, degree, and multiplicity to sketch graphs and describe end behavior. They begin by noticing patterns between the degree of a polynomial and the shape of its graphs, learning precise language for describing what happens to the function as the inputs get larger and larger in either the positive or negative direction. Students continue to notice patterns as they explore how the sign of the leading coefficient and the multiplicity of factors affect the graph of the polynomial. Students put all of these considerations together as they sketch graphs of polynomials written in factored form.

Graph of a polynomial function on xy-plane. Horizontal axis, scale -10 to 8, by 2’s Vertical axis, scale -300 to 50, by 50’s. The function starts by decreasing through a root at (-3 comma 0) to a local minimum near (-1 comma -253). The function then increases through a y-intercept at (0 comma -192) to a root at (4 comma 0), then begins increasing.

Lastly, students build on previous work with systems of linear equations to solve systems of equations that include at least one quadratic equation. They use both graphing and algebraic strategies to find points of intersection for two polynomial functions.