Root(zero) of a polynomial

Question

How is the Root of a polynomial defined? I found on Wikipedia's page written: "A root of a polynomial is a zero of the corresponding polynomial function." Or the URL:https://en.m.wikipedia.org/wiki/Zero_of_a_function Now if I want to find the root of polynomial 9a-2b then that would be it's corresponding function?

Answer 1

If you are working over a field $F$, then the function will be$$\begin{array}{ccc}F\times F&\longrightarrow&F\\(a,b)&\mapsto&9a-2b.\end{array}$$

Answer 2

A root of a polynomial is the value (or values) given to its variable (or variables) that results in the polynomial evaluating to zero. For example, the root of a polynomial $x+2=0$ is $x=-2$. Likewise, roots of the polynomial $x^2-1=0$ are $x=\pm 1$. In your case, the polynomial has two variables. But we have only one equation $9a-2b=0$. Therefore, the roots of this polynomial are any pair of numbers $a$ and $9a/2$.

Answer 3

$9a-2b$ is not a polynomial without any more context. It is a linear formula in two unknowns, so we could indeed see it as a polynomial in the two variables $a,b$, say $p(a,b) = 9a-2b$ (which also defines a function). Over the reals, or the rational numbers, this has many zeroes, e.g. $(1,4\frac{1}{2})$ is a root of the polynomial in $\mathbb{Q}^2$, because if we fill in the values in $p$ we get $0$.

If $b$ were some known number or parameter, then we could also see it as a polynomial in $a$: $q(a) = 9a-2b$, and then it has only one root, if $b$ is in some field: $a = \frac{2b}{9}$, which lies in the same field as $b$ provided $9$ has an inverse (it probably has, as I think you're working over real, complex or rational numbers?). In this case $b$ is not seen as an "unknown" or variable that can be substituted.