Students identify an unknown coefficient in an expression for a polynomial function. While the link between seeing a factor to knowing that  is a zero of a function has been made in earlier lessons, this activity clarifies why this is true, using the relationships between division and multiplication equations.

Monitor for students who identify the value of  in these ways, ordered from more concrete to more abstract:

• Graphing for different values of to find a graph with as a horizontal intercept.
• Using long division and the knowledge that the remainder after dividing by is 0.
• Substituting 2 for and solving for when the expression is equal to 0.

Students using the first strategy may be relying on the pattern that a polynomial with as a factor intercepts the horizontal axis at and not thinking about the algebraic meaning of horizontal intercepts. Students using the second strategy, long division, may be remembering that if you divide by a factor, you get a remainder of 0. Only students using the third strategy are explicitly setting equal to 0 and connecting their understanding of linear factors of a polynomial, the zeros of a function, and function notation.

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 .)

The purpose of this task is to establish the Remainder Theorem and, from it, conclude that for a polynomial , if , then is a factor of . Students repeatedly perform polynomial long division problems to generalize the value of when is divided by (MP8), and so graphing technology is not an appropriate tool. Instead, encourage students to think of other strategies they can use to answer the questions.

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.