What does "does not cancel on the interval" mean? I have a question which asks me to consider the system
$$f(x)=a+bx+\frac{P(x)}{Q(x)},\qquad x\in I,$$
where $P$ and $Q$ are polynomials, and stipulates &quot;$Q(x)$ does not cancel on the interval $I$ &quot;.
What does this mean?
Does this mean $Q(x)\neq0$ on this interval, or that $Q(x)\neq P(x)$ on this interval, or perhaps something else? I have tried Googling the expression but couldn't find anything, and I haven't came across it before.
It might mean that whenever $Q(x)=0$, then $P(x)\ne 0$. And the other way around, $P(x)=0\implies Q(x)=0$, but that is probably the less interesting case.
As $P,Q$ are polynomials, the usual use of &quot;cancel&quot; is that a common factor can be removed, here that could be interpreted the way that $P,Q$ do not have a common factor that has roots inside that interval.
But it might also be an unfortunate word choice and the condition to express was that &quot;$Q$ does not vanish on $I$&quot;, which indeed then means that there are no roots of $Q$ in $I$.
It means $Q(x) \neq 0$ for all $x\in I$.