The purpose of this Warm-up is to focus student attention on the link between the divisor, dividend, quotient, and remainder and how those 4 terms can be used to write a multiplication equation whether they involve numbers or polynomials. This will be useful when students work with dividing polynomials and the meaning of the remainder in a later activity. While students may notice and wonder many things about these images, the idea that not all division works out “evenly,” along with the meaning of the remainder, are the important discussion points.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the difference between zero and non-zero remainders does not come up during the conversation, ask students to discuss this idea.

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.

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.