In an earlier activity, students graphed the function describing the relationship between the volume of a box made from a single piece of paper and the side length of the squares cut from the corners of the piece of paper. In this activity, students identify features of a similar polynomial function from the context and expression without a graph, so technology is not an appropriate tool. During the discussion, students use two equivalent forms of the expression to identify features of the graph, including intercepts.

Monitor for students who use these different strategies when determining the degree and leading term of the polynomial in the first question:

• Rewrite the equation in standard form. Some students may complete the multiplication in different ways. For example, given three factors , , and , some students may multiply , while others choose .
• Use diagrams to organize their work.
• Multiply  .

If students are unsure how to identify the degree or leading term of the polynomial written in factored form, consider asking:

• “What did you notice and wonder about Equation A in the previous activity?”
• “How could using the distributive property help you to identify the degree or leading term of the polynomial?”

The goal of this discussion is for students to consider the advantages and disadvantages of polynomials written in factored form and standard form, and to see different ways to organize the use of the distributive property.

Display the function  from the Task Statement for all to see. Invite previously selected students to share how they determined the degree and leading term of the polynomial. Here are some possible questions for discussion:

• "Does it matter the order in which the factors are multiplied?" (No. Multiplying the factors in any order will have the same result.)
• "How can drawing a diagram help to organize the work?" (A diagram can help make sure no terms were missed and help keep track of the signs of each term.) 

If not brought up in students' explanations, display the function  for all to see. Ask students which form of the polynomial, factored form or standard form, makes it easier to identify the following features:

• Degree
• Leading term
• Constant
• Horizontal intercept(s)
• Vertical intercept

If time allows, display a set of scaled axes and add these features to the display. Then sketch in the cubic curve of at the conclusion as a small preview of the work students will do in future lessons investigating the general shape of polynomials with different degrees.

Invite students to share their diagrams. If some students used methods other than diagrams to multiply the expressions, invite them to share and explain their method.

In this activity, students graph two related cubic polynomials to investigate the effect that multiplying by a constant has on input-output pairs of a function. Students are invited to use precise mathematical language as they describe how the features of the graphs are similar and different (MP6). They use two equivalent forms, standard and factored, to find properties of the functions, such as relative minimums and relative maximums.

In the digital version of the activity, students use an applet to visualize the effect of multiplying a polynomial by a constant, . The applet allows students to experiment with different values of . This activity works best when each student has access to devices that can run the Desmos applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, displaying the applet would be helpful during the Activity Synthesis. 

The purpose of this discussion is for students to understand that multiplying by a constant causes all output values of a function to be times farther from the horizontal axis, which in turn means has no effect on the zeros of a function.

Begin the discussion by selecting 2–3 students to share how they rewrote and in standard form. If necessary, remind students how multiplying in a different order results in the same product.

Ask students, “How can you identify the constant term without multiplying out the entire expression?” (With an expression like , the constant term is 21 and comes from .)

If not brought up during the discussion, it is important to note that all three polynomials have the same degree and the same zeros, yet the three have distinct outputs for all other inputs. The idea that knowing the degree and zeros of a polynomial is insufficient for identifying a specific polynomial will be revisited in future lessons. If time allows, ask students to write an equation for a fourth function with the same degree and zeros as the first three.