Study Guide
Learning Objectives
- Identify zeros and connect real zeros to linear factors and $x$-intercepts.
- Use multiplicity; know degree $n$ means $n$ complex zeros counting multiplicities.
- Apply conjugate pairs for real-coefficient polynomials.
- Relate even/odd multiplicity to tangent vs crossing behavior.
- Determine degree from successive differences on equal-interval tables.
Suggested Practice Sequence
1. Zero ↔ factor ↔ intercept translations. 2. Multiplicity sketches (1, 2, 3). 3. Degree accounting with mixed real/non-real zeros. 4. Build polynomials from zero lists (real coefficients). 5. Successive-difference degree from tables. 6. Mixed FRQ write-ups with 1.4 extrema language when relevant.
Classroom / Check-In Moves
- Require conjugate in the same sentence as any non-real zero.
- Have students annotate graphs "cross" or "touch" with multiplicity.
- Live table: compute differences until constant; announce degree.
Misconception Watch-List
- Non-real zeros are optional extras beyond the degree.
- Even multiplicity means the zero is not real.
- Successive differences work with any uneven $x$-list.
Vocabulary to Own
- zero / root
- multiplicity
- complex conjugate pair
- $x$-intercept
- successive differences
Materials Map
- Practice: zeros, factors, multiplicity, conjugates, differences
- Word: applied intercept/touch-cross stories
- Mixed / multi-*: link to degree, extrema between zeros (1.4)
- Review & practice test: timed zero accounting
- Interactive: quick checks