The Remainder Theorem, Polynomial Expansion, and Tangent Lines

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