Invite previously selected students to share how they identified the value of . Sequence the discussion of the methods in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• "What were some advantages and disadvantages to the graphing strategy?" (Graphing technology made it easy to graph different equations, but guessing and checking wasn't very efficient.)
• "Consider the division problem and the related multiplication equation from the Warm-up. How can we write a related multiplication equation for this division problem?" (, where is a cubic function.)
• “If you know that is a factor of the polynomial, how do you know that ?" (Because .)

If students do not yet feel fluent with polynomial division, consider asking:

• “How did you decide if is a factor?”
• “How could using graphing technology help you see which polynomials have as a factor?”

The goal of this discussion is to make sure students understand the results of the last question—the remainders when dividing the list of polynomials by are all the same as the value of the polynomials at . Begin by discussing:

• “What is the remainder when is divided by ?” (5)
• “What is the value of ?” (5)
• “What do you notice about polynomial functions that do not have a factor ?” (They all have a remainder when divided by . The remainder is the same as the value of the function when .)

Then ask students, “Why does it seem like the remainder when is divided by is equal to ?” Give students 2–3 minutes of work time. If needed, directly encourage students to use the relationship between division and multiplication that they have seen in the activities leading up to this one. Invite students to explain what they think is happening.

It is important for students to understand that since all division problems can be rewritten as multiplication problems, we can think of dividing the polynomial by the linear factor as , where is the remainder and is a polynomial. When , , so the remainder after division by is . Tell students that this is called the Remainder Theorem.

This theorem allows us to state that if we have a polynomial with a known zero at , then is a factor of since if we divide by , then we have , where is the remainder and is a polynomial. Because we already know is a zero of the function, we know that . This means we also know that the remainder is 0, since . So if is a zero of the polynomial.