Common Mistakes
Frequent Errors
1. Forgetting the conjugate when given one non-real zero. 2. Not counting multiplicity toward the $n$ zeros of a degree-$n$ polynomial. 3. Claiming the graph crosses at an even-multiplicity real zero. 4. Thinking non-real zeros appear on the $xy$-graph as intercepts. 5. Reading successive differences from unequal input spacing. 6. Confusing "complex zeros" (all zeros in $\mathbb{C}$) with "non-real only." 7. Writing factor $(x+a)$ when the zero is $a$ (sign error). 8. Assuming every polynomial's coefficients are real when applying conjugate pairs.
Why These Happen
Students meet zeros, factors, graphs, and complex numbers in one topic. Conjugates feel optional; multiplicity feels like extra credit; tables invite sloppy spacing.
How to Fix Them
- Habit: non-real zero → immediately write conjugate.
- Habit: add multiplicities; check against degree.
- Habit: even multiplicity → touch; odd → cross.
- For tables: verify $\Delta x$ constant before differences.
Avoidance Scripts
- "Pair first, then fill the remaining real zeros."
- "Multiplicities are the degree budget."
- "Touch = even; cross = odd."
Quick Self-Check
- Did I write both conjugates?
- Do multiplicities sum to degree?
- Does my sketch match multiplicity parity?