Zeros of Rational Functions

A rational function r(x) = p(x)/q(x) outputs zero exactly when the fraction equals zero, and a fraction equals zero only when its top is zero while its bottom is not. Dividing by zero never gives zero — it gives an undefined value. So the hunt for zeros starts and ends with the numerator: factor p(x), set each factor to zero, and those x-values are candidates.

The catch is that candidates can get disqualified. If a factor shows up in both the numerator and the denominator, it cancels algebraically, but the original function is still undefined at that x-value — it just shows up as a hole in the graph rather than a crossing through the x-axis. So after factoring both top and bottom, cross out any shared factor before you read off the zeros. What survives in the numerator after cancellation tells you the real zeros; what's left in the denominator tells you the vertical asymptotes.

This sits right after graphing behavior near asymptotes and holes, and right before end behavior from the leading terms. Zeros, holes, and asymptotes together give the full skeleton of a rational function's graph: where it touches down, where it breaks, and where it's simply missing a point. Getting the zero condition right — numerator zero, denominator nonzero — is what keeps all three pieces straight.