Abstract
In all we have done so far concerning polynomials, at no moment we had any intent of substituting X by an element of \(\mathbb K\). Even in Theorem 14.10 and Example 14.15, the notation f(z), used to denote the complex number obtained by formally substituting X by z in the expression of \(f\in \mathbb C[X]\), was a mere convention. This is no surprise, for we are looking at polynomials as formal expressions, rather than as functions. In this sense, the indeterminate X is a symbol with no arithmetic meaning, and we have even stressed before that we could have used the symbol \(\square \), instead.
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Notes
- 1.
Paolo Ruffini, Italian mathematician, and William G. Horner, English mathematician, both of the eighteenth and nineteenth centuries.
- 2.
- 3.
For other approaches to this problem, see Problem 4, page 444, and Problem 3, page 450.
- 4.
The reader acquainted with Calculus has certainly noticed that the definition of f′ matches the one presented in Calculus courses by computing limits of Newton’s quotients. The point of the present definition and the subsequent proposition is that they will equally apply to polynomials over \(\mathbb Z_p\), in Chap. 19.
- 5.
Brook Taylor , English mathematician of the eighteenth century.
- 6.
One can show (cf. Problem 5, page 444) that, given a simple n-sided polygon, one can always find a polynomial function of degree at most n − 1 and whose graph contains the set of vertices of the polygon.
References
A. Caminha, An Excursion Through Elementary Mathematics II - Euclidean Geometry (Springer, New York, 2018)
C.R. Hadlock, Field Theory and its Classical Problems (Washington, MAA, 2000)
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Caminha Muniz Neto, A. (2018). Roots of Polynomials. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_15
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