Why are so many people under the impression that we need to find the domain of a rational function in order to find its zeros? Zeros of #f(x) = (x^2-x)/(3x^4+4x^3-7x+9)# are #0,1#.
I think that finding the domain of a rational function isn't necessarily related to finding its roots/zeros. Finding the domain simply means finding the preconditions for the mere existence of the rational function.
In other words, before finding its roots, we have to make sure under what conditions the function does exist. It might seem pedantic to do so, but there are particular cases when this matters.
My guess is that a factor in the numerator could also be represented in the denominator, resulting in a removable discontinuity.
This is just my speculation, but I'd bet the problem occurs with finding the zeros of a function like this:
You'd be tempted to say the zeros are at
If you factor the denominator (and numerator), you get
So the function is really just
This also could apply to functions with odder denominators. I really don't think this is incredibly important to note, since it's rare this is ever an issue, but in
The domain doesn't include
So in a function like
There's not a zero at