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AP Calculus Question of the Month: September 2003
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by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut
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Is There Anything Remaining from the Remainder Theorem?
The Remainder Theorem says that if we divide a polynomial 
with real coefficients by a linear divisor 
, the result is a quotient polynomial
of one less degree and a real remainder 
that satisfies 
. In other words, the remainder is the numeric value of the polynomial at 
. Its corollary, the Factor Theorem, says that 
if and only if 
is a factor of 
. So graphs of polynomials link factors to x-intercepts.
Before graphing calculators, synthetic division was the usual way to find quotients and remainders for division of polynomials. Now graphing calculators with Computer Algebra Systems can do the job. Use the proper fraction operation (propFrac) to find the quotient and remainder. The figure shows the calculator output when we divide the polynomial 
by 
. The third line gives the quotient polynomial 
and the remainder 
. Note 
in the fourth line. The quotient polynomial evaluated at 
coincides with 
in the fifth and sixth lines.
The last two lines of output suggest there is calculus here too. Use your calculator to graph 
and 
in the vicinity of 
The graphs should intersect at 
For this month's question, let's generalize this interesting behavior.
Prove that 
by
(A) Using the definition of derivative, and
(B) Using one, or more, of the formulas for finding derivatives.
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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