a n 1 a n 1 a × 1 a × n t i m e s For example, 4 -3 Here, the base number is 4 and the exponent is -3. I have no hard evidence of this so take it worth a grain of salt, but it would be more consistent with your suggested definition. Negative Exponent Rule 1: For every number a with negative exponents -n (i.e.) a -n, take the reciprocal of the base number and multiply the value according to the value of the exponent number. There is one quotation from there that suggests to me the original meaning of polynomial may have been: Any string of mathematical expressions connected by addition and subtraction. There is a brief investigation of the origin of the word "Polynomial" here : Link Then when the student is comfortable with polynomials adding the more complicated negative powers is just an easy modification of what has already been learned. Polynomials are probably the most well behaved functions there are (no singularities, continuous and differentiable everywhere, etc.) it is helpful to study them separately first. There are a lot of rules that apply to the polynomial family as a whole which you would lose if you included negative powers into the definition. There are good pedagogical reasons to teach polynomials with just positive powers. Laurent Series), but then you just wouldn't call it a polynomial. Practically speaking you frequently want to work with negative powers of $x$ as well as positive powers (e.g. Polynomial is just a name for a certain kind of structure. I don't know if there is a "good" reason. When both factors are in the numerator, the remaining denominator is a factor of \(1\) which generally is not written. There are, indeed, other algebraic structures that allow the evaluation functions to constants, but the ring of real polynomials $R$ is the only one that allows the variable $x$ to be evaluated to an element of any larger structure containing $\mathbb R$ (to any $R$-algebra). In this expression, both \(x\) and \(y\) will have negative exponents when they are in the numerator and positive exponents in the denominator. If there is something more complex such as then only the x will go to the denominator since it is the one with the negative exponent. Negative exponents are a way of writing powers of fractions or decimals without using a fraction or decimal. I stress that the domain is the set of polynomials, and the target space, also known as codomain, is $\mathbb R$ (the algebra $A$ in the general case). so the final answer is To simplify the following: we have to move the x to the denominator and make the exponent positive. The function $e_a$ may be called the evaluation morphism. That is, it satisfies $e_a(f g)=e_a(f) e_a(g)$ and $e_a(fg)=e_a(f)e_a(g)$, as well as $e_a(\mathbf1)=1$, where the bold-face $\mathbf1$ is the constant polynomial $1$, and the other $1$ is the ordinary unit element of $\mathbb R$. And this function is both additive and multiplicative (it’s a ring morphism from $R$ to $A$). That is, once you’ve chosen $a$, then $f(a)$ makes sense as a real number (as an element of $A$) no matter what polynomial $\,f$ you look at.įurthermore, the choice of $a$, once it’s done, gives you a function from polynomials to constants, I’ll call it $e_a$, namely $e_a(f)=f(a)$. Just to speak only of polynomials in one variable, the set of all such, $\mathbb R$ ($R$ for a general ring), has the property that the variable $x$ may be evaluated to any real number $a$ (to any element $\alpha$ of an algebra $A$ over the base ring $R$) so that this “evaluation mapping” can be applied to any polynomial at all (to any element of $R$ at all). The set (ring, actually) of polynomials with real coefficients (more generally with coefficients in any commutative ring) has a “universal” property that the larger sets do not. I’ll try to make my explanation as elementary as I can, with parenthetical expansions for the more technically inclined. Here’s my understanding of the reason for the restrictiveness of the definition of “polynomial”, in response to suggestion. I would opine that the reason that they're not the primary object of study is because they're not the 'simplest' structure of interest among any of their peers, and fundamentally the most important structures in mathematics tend to be the simplest structures exhibiting some given property. As that article suggests, they're of particular importance and interest for their connections with the field of Hopf Algebras (and by extension, quantum groups). What's more, there's also an object occasionally studied that more directly corresponds to your notion: the notion of Laurent Polynomial. \).Polynomials are defined as they are for a few distinct reasons: (1) because polynomials as functions have certain properties that your 'polynomials with division' don't have, and (2) because there are other terms for more generalized algebraic forms.įirst, the properties of polynomials: unlike e.g., $2x^$.
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