2025 AMC 12A Problems/Problem 25
Polynomials 
and 
each have degree 
and leading coefficient 
, and their roots are all elements of 
. The function 
has the property that there exist real numbers 
such that the set of all real numbers 
such that 
consists of the closed interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
together with the open interval 
. How many functions 
are possible?
Solution 1
From the given 
on ![$[a,b] \cup (c,d)$](http://latex-new.aopstest.com/2/3/e/23ee6de2e2eabfa1fc76b5fa58eb4244b85aa39e.png)
and 
elsewhere, we deduce:
and 
are zeros of 
(since we transition from 
to 
at and from to at ).
and 
are poles (vertical asymptotes or holes) of since on 
we have , at 
and 
the sign is negative, and immediately outside the interval .
and 
are zeros of 
(since we transition from 
to 
at 
and 
are poles (vertical asymptotes or holes) of 
and 
the sign is negative, and immediately outside the interval Thus the sign pattern is:
- Victor Zhang
