# Root Preserving Transformations of Polynomials

by Branko Ćurgus (Western Washington University) and Vania Mascioni (Ball State Uniuversity)

Mathematics Magazine
April, 2007

Subject classification(s): Algebra and Number Theory | Linear Algebra | Linear Transformations

The article answers negatively the question, “Is there a (non-trivial) linear transformation $T$ from $P_n$, the vector space of all polynomials of degree at most $n$, to $P_n$ such that for each $p$ in $P_n$ with a real or complex root, the polynomials $p$ and $T( p)$ have a common root?" The proof is based on the fact polynomials of degree at most $n$ have at most $n$ roots in the real or complex numbers. This article investigates an area common to algebra and linear algebra.

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Capsule Course Topic(s):
Linear Algebra | Linear Transformation
Linear Algebra | Vector Spaces, Subspaces