Common Mistakes
Frequent Errors
1. Calling every numerator zero a zero of $r$. 2. Calling denominator zeros “zeros of the function.” 3. Forgetting domain when a factor cancels (hole preview). 4. Building inequality charts with only numerator zeros. 5. Mixing up $x$-intercepts with vertical asymptotes. 6. Solving $q(x)=0$ and thinking $r(x)=0$ there. 7. Dropping multiplicity information that affects sign charts. 8. Accepting calculator intercepts without checking excluded $x$.
Why These Happen
“Set numerator = 0” is memorable; the domain filter is not. Inequality work feels like polynomial inequalities, so students forget denominator barriers. Holes and asymptotes blur together early.
How to Fix Them
- Two-column habit: candidates | excluded.
- Say out loud: “Zero of $r$ means $r(a)=0$, so $q(a)\ne 0$.”
- Inequality: mark num zeros (filled/open per inequality) and den zeros (never solutions; asymptote/hole barriers).
Avoidance Scripts
- “Numerator proposes; domain disposes.”
- “Den zero: not a zero of $r$.”
- “Inequalities need both polynomials’ real zeros.”
Quick Self-Check
- Did every listed zero satisfy $r(a)=0$?
- Did I exclude den zeros?
- For inequalities, are den zeros on the chart?