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AP Calculus Question of the Month: November
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by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut
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Fun Investigations with a CAS: Third Degree Polynomials
Polynomials have several interesting properties, especially those of degree three. Unfortunately, a lot of symbolic manipulation is required to prove these properties, but a computer algebra system (CAS) can really lend a hand. A CAS will perform any algebra and calculus operations you specify. We begin with this exploration of one property of cubic polynomials. This month's problem is proposed on two levels: first determine how to prove the property, and then determine how to use a CAS to complete the proof in as few lines of code as possible. Here is this month's question:
Prove that the tangent line drawn to a cubic polynomial at the point where x = average of two of its roots, intersects the polynomial on the x-axis at the third root.
Once you have a solution mapped out, try to minimize the computations: par for a CAS is three lines of computation, but birdies are possible. I suggest you begin with the case of three distinct real roots. Don't overlook the cases of a triple real root or a double real root (and another real root). Notice that this problem only arises if there are three real roots, not one real root (and two complex roots).
Complete the question before viewing the answers and explanation!
Lin McMullin, an educational consultant and writer with extensive experience teaching AP Calculus, lives in Niantic, Connecticut. As a College Board consultant he has presented AP Calculus institutes and workshops in the United States and Europe, and is a Table Leader at the Reading. His work as a writer includes co-authorship of the popular D&S Marketing Systems review books for the AP Calculus Exam and Teaching AP®
Calculus, a book especially for AP Calculus teachers.
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