The goal of this activity is for students to understand how a diagram can be used to organize dividing polynomials. This activity continues an idea started earlier—if is a zero of a polynomial function, is a factor of the expression?

After students work out what quadratic times is equivalent to the original expression. They factor the quadratic using a method of their choosing and then make a sketch of the 3rd-degree polynomial.

If students are unsure of how to complete the diagram, consider asking:

• “Tell me more about your diagram.”
• “How could you use the examples from the Warm-up to figure out where each term comes from and determine an order to fill in the parts of Priya’s diagram?”

The goal of this discussion is for students to share how they worked out the terms of the quadratic factor of . It is important for students to understand that it isn't until the value in the bottom right box of the diagram works out to be -8 that we know for a fact that is a factor. Here are some questions for discussion:

• “How did you figure out what term must be above ? Below it?” (Since , I knew must be above , and that means is below it, since .)
• “How did you work out the next column?” (The expression for has , so with already known, the term next to must be , making the term above it , since , and the term below must be , since .)
• “If the last term in the lower right hadn’t been -8, what would that mean?” (That isn’t a factor of .)

Students practice using a diagram to factor when one factor (or more) is already known. The first few questions have diagrams started to give students more examples to study (in particular, that 0 is an acceptable entry). Students use repeated reasoning (MP8) as they work through the logic of polynomial division to identify other factors, so technology is not an appropriate tool.

Students will need to make additional diagrams for the last two problems. If students start with the original polynomial instead of the result of the previous division, consider asking:

• “Can you explain how you rewrote your polynomials.”
• “If you want to know the prime factors of 60, you can divide 60 by 5 to get 12. Then you would divide 12, not 60, by another number to continue the factorization. How could a similar process work when finding factors of polynomials?" 

The purpose of this discussion is for students to see and hear the ways their classmates approached the questions in the activity. Invite 1–2 students per question to share their diagram and how they worked out the other linear factors of the original expression. If time permits, highlight that for  whether we divide by  first or by  first, the result is still that .