Why must polynomials have degrees that are positive integers?

Question

What's the benefit of knowing 'this' is a polynomial? And why should we care?

Answer 1

Polynomials are everywhere infinitely differentiable (not to mention defined and continuous!), and positive-degree polynomials in one variable are completely factorisable into linear factors over $\Bbb C$. If we allow negative powers, the example $\frac{1}{x}$ is undefined at $0$ and has no roots in $\Bbb C$. Thus polynomials as we currently define them are a very nice set of functions to consider.

Answer 2

Polynomials are obtained by means of addition and multiplication alone. That gives them many interesting properties such as continuity, differentiability, null $d+1^{th}$ derivative, ring algebra, known number of roots, connection to linear algebra via ODE's and Eigenvalues and many many more.

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Generalized polynomials with negative integer powers can be rewritten as the ratio of a polynomial over a power and are of little interest ($x+4x^{-3}=\dfrac{x^4+4}{x^3}$). Rational fractions, i.e. the ratio of two polynomials, are richer and more important.

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Generalized polynomials with rational exponents can also be turned to ordinary polynomials by a change of variable (e.g. $x^{3/2}+4x^{1/3}=y^9+4y^2$ with $x=y^6$). So nothing really new.

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Finally, generalized polynomials with real exponents are not attractive. They cannot be defined on negative arguments (and fit poorly with complex numbers) and don't enjoy the composition property "$p(q(x))$ is a polynomial". E.g.

$$(x+1)^\pi$$ cannot be written as a generalized polynomial in $x$.