Education Technology

The Remainder Theorem, Polynomial Expansion, and Tangent Lines

Published on 07/20/2005

Activity Overview

In this Derive™ activity, students use the Remainder Theorem to find a tangent line to the graph of a polynomial function.

Before the Activity

  • See the attached DFW file for detailed instructions for this activity
  • Print pages from the attached DFW file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity Procedures:

  • Set up a 2D plot window
  • Graph the given polynomial
  • Plot the equation whose right side is formed by constant term and the linear term of the polynomial in the same window
  • Note that the graph of the line is tangent to the graph of the polynomial at x = 0
  • When x is close to 0, the constant term plus the linear term dominate the value of the polynomial
  • Subtract these terms from the polynomial, set the window range, and graph the result
  • Note that the graph of the resulting polynomial lies entirely within the set window
  • Understand that the sum of the constant and linear terms will be almost equal to the value of the polynomial for a given x-value in an interval
  • Understand that lower degree terms dominate the polynomial
  • Realize that when x lies between -1 and 1, the dominance of x increases with its powers


  • Study the statement of the Remainder Theorem
  • Use the Derive steps to simplify the expression for the polynomial
  • Observe that the result is another constant polynomial
  • Evaluate it to obtain the right hand side of the given expression in the statement of the Remainder Theorem
  • Recognize the fact that if P(x) is a polynomial, then the tangent line to the graph of P(x) at x = a is given by the expression Q(a)(x - a) + P(a), where Q(a) is the polynomial (P(x) - P(a))/(x - a)
  • After the Activity

    Students answer questions and solve problems listed on the activity sheet.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary