Using MATHEMATICA to Prove and Animate a Property of Cubic Polynomials

Alan Horwitz

Department of Mathematics
Marshall University
Huntington, WV 25755
USA
Phone: (304) 696-3046
Horwitz@Marshall.edu

list of all papers by this author

Click to access this paper: paper.pdf

ABSTRACT

In this paper, we use the capabilities of Mathematica for manipulating algebraic expressions to prove the following proposition:

For any cubic polynomial p(x) with real roots r1, r2 and r3, the tangent line to the graph of y = p(x) at x = ( r1 + r2 )/2 always has ( r3 , 0 ) as its x- intercept.

In addition, we explain how Mathematica can be used to animate the graph of y = p(x) as one root varies and the other two roots remain fixed, in order to demonstrate that the tangent line, described above, behaves in the same way for any choice of real roots. To generate each frame of the animation, we use a program which, for any function f : R ---> R, draws both the graph of f and a tangent line to the graph at a location halfway between any chosen pair of zeros for f. We also generate pictures which show that the above proposition need not hold when f is chosen to be a non-cubic polynomial.

Keyword(s): Mathematica, precalculus