Polynomial and Sign Suppose I have a polynomial
$f(x)=0$ as $f(x)=1 + 2x - 3x^2 - 8x^3$ I can write this as $1+2x-3x^2 - 8x^3=0$. Suppose I want to write the coefficient of the highest power in positive term, so I write the equation as $8x^3+3x^2-2x-1=0$. Can I still call this $f(x)$ or will this be $-f(x)$.
Edit:
Context. I need to find the asymptotes for the curve $x^3+2x^2y-4xy^2-8y^3+4x-8y=1$, I form $\frac{c^2}{2}\phi^{’’}_{3}(m)+c\phi^{’}_{2}(m)+\phi_{1}(m)=0——-(1)$ Where $\phi_{3}(m)$ is the  collected terms of highest powers which is 3 here where we put $x=1, y=m$. My question is in this equation $\phi_{3}(m)=2+2m-4m^2-8m^3$ but in equation (1) if put in the place of $\phi_{3}(m), 8m^3+4m^2-2m-1$  the asymptotes have complex term. Is this valid? In one text book they have changed the sign for $\phi_{3}(m)$ but not for $\phi_{1}(m)$ and obtain real values. Can we be inconsistent like this about the sign of the polynomial? How does it affect the result here?
Also can someone point out the theoretical background for this method? This particular text book doesn’t give any justification or motivation for this method.
 A: These are completely different function. And if you use the same $f(x)$ to reference different functions, it's going to be confusing.
For some procedures this won't matter. E.g. both $f(x)$ and $-f(x)$ have the same roots. So if it's easier to find roots of $-f(x)$, then we can switch to working with that function for time being. But it doesn't make them equal on all $x$. We can give it a different name like $g(x) = -f(x)$.
Here's another situation:
$$
f(x)=\frac{4x(x-1)}{x-1} \ne 4x
$$
These functions are even more similar as they are equal on almost all points. Except for $x=1$, where the original one doesn't have an answer. But since they are different at least one $x$, then we can't say it's the same function.
Now suppose we write it like this:
$$
f(x)=\frac{4x(x-1)}{x-1} = 4x \text{, for } x\ne1
$$
This way the functions become completely equivalent. Now you have a choice to throw away the original definition of the function and override it with the new one; or keep them separately and give them separate names. E.g. in situations when you want to reason about the difference in the calculation procedures between them you may as well give them different names like $f(x)$ and $s(x)$.
So whether you can re-use the name will depend on the context. But strictly speaking if the functions look different, they are different functions. Even if they are just different ways of calculating the same thing. But in some situations you'd like to throw away the original function in favour of the new one and want to re-use its symbol.
