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Zeroes and factorization of a non polynomial function
Task
 Sketch graphs of the functions $f$ and $F$ given by $f(x)= x$ and $F(x) = x^2$ for $2 \leq x \leq 2$.
 Suppose $g$ is the function given by $g(x) = \frac{f(x)}{x}$ for $x \neq 0$ and $G$ is the function given by $G(x) = \frac{F(x)}{x}$ for $x \neq 0$. Sketch graphs of the functions $g$ and $G$ for $x \neq 0$ and $2 \leq x \leq 2$.
 Is there a natural way to define $g$ and $G$ when $x =0$? Explain.
IM Commentary
For a polynomial function $f$, if $f(0) = 0$ then the polynomial $f(x)$ is divisible by $x$. This fact is shown and then generalized in ''Zeroes of a quadratic polynomial I, II'' and ''Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient $\frac{f(x)}{x}$ will still be a nice function  indeed, another polynomial, save for the missing point at $x=0$. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The nonpolynomial function $F$ given by $F(x)=x$ is a familiar function for which property does not hold: even though $F(0)=0$, the quotient $\frac{F(x)}{x}$ behaves badly near $x=0$. Indeed, its graph is broken into two parts which do not connect at $x = 0$.
The level of the task is appropriate for assessment but since its intention is to provide extra depth to the standard AAPR.2 it is principally designed for instructional purposes only. The students may use graphing technology: the focus, however, should be on what happens to the function $g$ when $x = 0$ and the calculator may or may not be of help here (depending on how sophisticated it is!).
Solution

Below is a picture of the graph of the equation $y = x$ when $2 \leq x \leq 2$: it is the graph of the equation $y = x$ when $0 \leq x \leq 2$ and the graph of the equation $y = x$ when $2 \leq x \leq 0$.
Below is a picture of the graph of the equation $y = x^2$ when $2 \leq x \leq 2$:

If $g$ satisfies $g(x) = \frac{f(x)}{x}$ this means that $g(x) = \frac{x}{x}$. When $x \gt 0$ we have $x = x$ and so in this case $$ g(x) = \frac{x}{x} = 1. $$ Similarly, if $x \lt 0$ then $x = x$ and so $$ g(x) = \frac{x}{x} = 1. $$ Below is a graph of the function $y = g(x)$:
If $G$ is a function which satisfies $G(x) = \frac{F(x)}{x}$ this means that $G(x)= \frac{x^2}{x} = x$. The graph of $G$ is shown below for $2 \leq x \leq 2$ and $x \neq 0$:

The graph of $g$ is broken into two parts which do not come together when $x = 0$. To the right of $x = 0$, $g$ always takes the value $1$ while to the left of $x = 0$, $g$ always takes the value $1$. Coming from the left $g$ appears as if it will take the value $1$ when $x = 0$ while coming from the right it appears as if $g$ will take the value $1$ when $x = 0$. There is no ''natural'' choice for what value $g$ should take when $x=0$.
Unlike $g(x)$, the function $G(x)$ looks exactly like the function $h(x) = x$ except that the point $(0,0)$ is missing. This makes sense because $$ G(x) = \frac{x^2}{x} = x $$ and the only problem with $G$ is that it has been expressed with an $x$ in the denominator so that it is not defined when $x = 0$. Looking at the graph of $G$, however, it is natural to define $G(0) = 0$.
Zeroes and factorization of a non polynomial function
 Sketch graphs of the functions $f$ and $F$ given by $f(x)= x$ and $F(x) = x^2$ for $2 \leq x \leq 2$.
 Suppose $g$ is the function given by $g(x) = \frac{f(x)}{x}$ for $x \neq 0$ and $G$ is the function given by $G(x) = \frac{F(x)}{x}$ for $x \neq 0$. Sketch graphs of the functions $g$ and $G$ for $x \neq 0$ and $2 \leq x \leq 2$.
 Is there a natural way to define $g$ and $G$ when $x =0$? Explain.
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