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What is the rationale that the degree of a polynomial is non-negative? Can the degree be a fractional number. why the definition is only with the non-negative integers

We are bounded by the definition But I am curious to know the reason

Achari S Ganesha
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3 Answers3

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A polynomial of the form $p(x)=a_0+a_1x+...+a_nx^n$ has a degree $n\in\{1,2,3,...\}$. This definition allows polynomials to be continuous and differentiable on $\mathbb{R}$. If thew degree is a fraction or a negative number then we end up with different kinds of functions that are not continuous in $\mathbb{R}.$ If the degree is $1/2$, for instance, then we could have functions of the form $$f(x)=(2x+3)^{1/2}=\sqrt{3x+3}$$ which we know has a restrcited domain. Also, if the degree is $-2$, then we may have a function of the form $$f(x)=(2x+3)^{-2}=\frac{1}{(2x+3)^2}$$ which we know is also not continuous on $\mathbb{R}.$

There may be more technical\potentially philosophical reasons, but I am not familiar with them as yet.

user138999
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  • Please write how can one explain this to school children – Achari S Ganesha Jul 10 '14 at 11:42
  • Hmm. If the children are old enough or rather fluent enough at mathematics to understand what a polynomial is then they should be able to understand this. Simply put: a polynomial has coefficients that are elements of the natural numbers because if this were not the case, then such a function would not be a polynomial. That is rather circular logic but I cannot put it simpler than that. – user138999 Jul 10 '14 at 11:45
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Please read carefully the definition of polynomial on Wikipedia: a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

W. Robin
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A polynomial is an algebraic object. You take any domain $R$ of coefficients, which should be at least a (commutative, abelian) ring, and introduce an external element $x$. With this external element you also add all combinations that result from the formal application of the ring operations, addition, subtraction and multiplication. The result is the polynomial ring $R[x]$, and $x$, by construction, only occurs in integer powers.

For a reasonable introduction of rational powers you have two ways (possibly more): You can consider the algebraic closure of $R[x]$, i.e., solutions $y$ of polynomial equations $f(x,y)=0$. Or the algebraic closure of the power series ring $R[[x]]$, the ring of Puisseux series. There you always have a common denominator in all the powers occurring in each series, and a finite number of negative powers may be present.

Lutz Lehmann
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