# Activities

• • • ##### Subject Area

• Math: Algebra II: Polynomials

• ##### Author 9-12

45 Minutes

Derive™ 6

• ##### Report an Issue

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.

• 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