Reflection: Rational Roots and Remainders: Important Theorems of Polynomials - Section 2: Investigation: Remainder Theorem


All my students completed the tables perfectly. But most failed to make the full connection and master the learning target in the observation question. Here are some sample student responses for the observation section of this investigation:

  • Some students simply missed the mark and did not demonstrate mastery:

         “The opposite of the divisor equals the remainder.”  - very common answer!

         “Almost the same remainder."


  • Many students got that the remainder was the same as the function being evaluated at that same value of k. But I don’t think they understand what k was or how k was related to the problem in first place. So most students actually only showed a partial mastery:

         “The remainder of the first box and the answer of the second box are the same.”

         “I observe that the remainders and f(k) are the same exact answers.”

         “The remainders are the same.”

         “The remainder is related to f(k).”

         “If you substitute the k values in the function you get the remainder.”

         “It’s just backwards. Remainders are the same as f(k).”

Many students were not precise in documenting their observations. A majority of my students fell into the 3rd category above, so it make me think that the “what did you observe?” question should be more direct. Or maybe I should add a question about “what is k? Or how can we find k?” I would really like students to make this connection on their own without me walking them through it though. In the future, I will monitor student work more closely on this and ask students to be more precise in their answers by asking questions like “What do the values in the table that you found represent? What is their relationship? This is true when? What is k? How can we find it?”

  Student Work – More practice needed on Mathematical Practice 6!
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Rational Roots and Remainders: Important Theorems of Polynomials

Unit 2: Polynomial Functions and Equations
Lesson 9 of 15

Objective: SWBAT identify the connections between dividing polynomials and evaluating polynomials and determine the possible rational zeros of a polynomial using the Rational Root Test.

Big Idea: Using polynomial division students discover the Remainder Theorem & then learn about the Rational Root Test.

  Print Lesson
Math, factor theorem, Precalculus and Calculus, polynomials, Algebra 2, PreCalculus, rational root test, rational zero theorem, Remainder Theorem
  51 minutes
polynomial zeros
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