Common Mistakes
Frequent Errors
1. Treating canceled rational expressions as identical with no $x\neq c$. 2. Reading zeros from standard form by guessing instead of factoring. 3. Using factored form alone for end behavior (missing leading-term analysis). 4. Setting the slant asymptote equal to the remainder. 5. Forgetting $\deg r < \deg g$ after division. 6. Multiplying back incorrectly to check $f=gq+r$. 7. Wrong Pascal coefficients or missing alternating signs for $(a-b)^n$. 8. Mixing hole language with division remainder language.
Why These Happen
Students memorize “cancel” and “FOIL” as goals instead of “choose the form that answers the question.” Slant asymptotes feel like “the leftover,” so remainder gets promoted to the asymptote.
How to Fix Them
- Label every rewrite: *equivalent for $x\neq \ldots$* when needed.
- Dual checklist: zeros? → factor. End behavior? → leading terms. Slant? → divide.
- Always multiply $gq+r$ to verify.
- Write Pascal’s row before expanding.
Avoidance Scripts
- “Same values where both defined; state restrictions.”
- “Quotient is the slant; remainder is not.”
- “Signs alternate when it’s $(a-b)^n$.”
Quick Self-Check
- Did I pick the form that matches the asked feature?
- Did I restrict the domain after canceling?
- Did I verify division by expanding $gq+r$?
- Are binomial signs correct?