Abstract
In this chapter we enrich our algebraic point of view of polynomials by considering them as functions. We develop first order analysis (critical points and monotonicity) for graphs of polynomial functions using synthetic division applied to difference quotients. We treat the difference quotient of a polynomial as a rational function with a removable singularity at the point where the quotient is taken. Removing the singularity then takes us directly to the concept of the derivative without taking limits. We discuss the special case of cubic polynomials in great details. In the second half of this chapter we return to algebra and study the roots of polynomials, once again with full details of the cubic case. We finish this chapter by the somewhat more advanced topic of multivariate factoring.
“In our days Scipione del Ferro of Bologna has solvedthe case of the cube and first power equal to a constant,a very elegant and admirable accomplishment. …In emulation of him, my friend Niccolò Tartaglia of Brescia,wanting not to be outdone, solved the same case when he gotinto a contest with his [Scipione’s] pupil, Antonio Maria Fior,and, moved by my many entreaties, gave it to me.”
in Ars Magna by Gerolamo Cardano (1501–1576)
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Notes
- 1.
A parabola with vertical symmetry axis is defined as the graph of the polynomial function y = ax 2 + bx + c, \(0\neq a,b,c\in \mathbb {R}\). See Section 8.2.
- 2.
Note the unfortunate double appearance of the symbol p. We will keep the polynomial p(x) and the coefficient p separate.
- 3.
Inspired by a problem in the Kettering University Mathematics Olympiad, 2007. Similar problems abound in mathematical contests.
- 4.
See the author’s Glimpses of Algebra and Geometry, 2nd ed. Springer, New York, 2002.
- 5.
The author is indebted to one of the reviewers for raising this question.
- 6.
This problem can be reformulated to finding the integer points on the elliptic curve y 2 = x 3 −x + 1 (with x = n − 1).
- 7.
This was a problem in the Canadian Mathematical Olympiad, 1971.
- 8.
The AC method is tedious and has very limited applicability. (It is unclear why this method plays such a paramount role in teaching basic algebra in schools.) Not only do the coefficients a, b, c have to be integers (or rational numbers at worst), but the AC method works if and only if the roots are rational numbers.
- 9.
As a nineteenth century mathematician would call it.
- 10.
When discussing gcf (a(x), b(x)), we always tacitly assume that at least one of the polynomials a(x) or b(x) is non-zero.
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Toth, G. (2021). Polynomial Functions. In: Elements of Mathematics. Undergraduate Texts in Mathematics(). Springer, Cham. https://doi.org/10.1007/978-3-030-75051-0_7
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DOI: https://doi.org/10.1007/978-3-030-75051-0_7
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