Is $p(x)=a\frac{x^2}{x}+b$ a linear polynomial? My question at first, may seem too simple, 


  Is $p(x)=a\frac{x^2}x+b$ a linear polynomial?


But as we see the polynomial $p(x)$ is not continuous at $x=0$. So if it is not a polynomial, which function is this?  
I got this definition on Wikipedia: 


  "Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc."
You are correct that it is NOT a polynomial. However it has a unique continuous extension to a polynomial ($ax+b$). However AS WRITTEN, it is not a polynomial.
Your question is philosophical haha. Really, the function is not defined in $x=0$. But... We can redefined f in $x=0$ because the limit exist! :).

In complex analysis this is a "singularity"