Study Guide
Learning Objectives
- Rewrite polynomial and rational expressions in equivalent forms (1.11.A).
- Use factored form to extract zeros, intercepts, asymptotes, holes, domain, and range.
- Use standard form to determine end behavior.
- Apply different representations to answer questions in context.
- Divide polynomials: $f(x)=g(x)q(x)+r(x)$ with $\deg r < \deg g$ (1.11.B).
- Use the quotient from long division to write slant asymptote equations.
- Expand $(a+b)^n$ with the binomial theorem / Pascal’s triangle (1.11.C).
Suggested Practice Sequence
1. Factor ↔ expand simple quadratics and cubics; name what each form reveals. 2. Rational cancel ↔ restricted equivalent expression; state domain. 3. Long division with remainder; rewrite $f=gq+r$. 4. Slant asymptote from quotient when $\deg(\mathrm{num})=\deg(\mathrm{den})+1$. 5. Binomial expansions $n=2,3,4$. 6. Mixed FRQ: given one form, answer with another.
Classroom / Check-In Moves
- Two-column: “Best form for ___?”
- Cold-call: after canceling, “What’s still restricted?”
- Require $f=gq+r$ written before claiming a slant asymptote.
Misconception Watch-List
- Canceling changes the function with no domain note.
- Slant asymptote is $y=$ remainder.
- Standard form automatically shows zeros.
- Binomial signs ignored when $(a-b)^n$.
Vocabulary to Own
- equivalent representations
- factored form / standard form
- polynomial long division; quotient; remainder
- slant (oblique) asymptote
- binomial theorem; Pascal’s triangle
Materials Map
- Practice: rewrite, divide, expand
- Word: context needing the “right” form
- Mixed / multi-*: combine with zeros, holes, end behavior, slant
- Review & practice test: timed representation fluency
- Interactive: quick checks