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BetterGrades Precalculus · Unit 5 · Unit map

Polynomial Functions

Power functions f(x)=ax^p connect positive, negative, and fractional exponent behavior.

Precalculus · Unit 5

The lesson path

Power functions f(x)=ax^p connect positive, negative, and fractional exponent behavior.

Core sequence

Polynomial Functions

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  1. 01Power functions and dominant behaviorCompare power functions f(x)=axpf(x)=ax^p and explain how exponent and coefficient control domain, symmetry, end behavior, and growth.
  2. 02Polynomial functions, degree, and leading termIdentify degree, leading term, coefficients, and standard form, and explain why the leading term governs large-input behavior.
  3. 03End behaviorDetermine polynomial end behavior from degree parity and leading coefficient and express it verbally, graphically, and with limit-style notation.
  4. 04Zeros, factors, and interceptsConnect polynomial zeros, linear factors, solutions, and x-intercepts, and distinguish exact from approximate zeros.
  5. 05Multiplicity and sign behaviorRelate the multiplicity of a zero to sign changes, crossing or touching behavior, and local flattening.
  6. 06Graph construction from structureConstruct a polynomial graph from degree, leading coefficient, zeros, multiplicities, sign intervals, and selected points.
  7. 07Turning points and local versus global behaviorUse degree bounds, graph evidence, and function values to analyze turning points and extrema.
  8. 08Intermediate Value reasoning and root boundsUse polynomial continuity and sign changes to guarantee roots and bracket them within intervals.
  9. 09Polynomial divisionDivide polynomials using long and synthetic division and verify the division algorithm P=DQ+RP=DQ+R.
  10. 10Remainder and factor theoremsUse P(c) as the remainder on division by x-c and test whether x-c is a factor.
  11. 11Rational-root candidates and root searchGenerate all possible rational zeros of an integer-coefficient polynomial and organize an efficient exact-and-numerical search.
  12. 12Complex zeros and the Fundamental Theorem of AlgebraCount polynomial zeros over the complex numbers, use conjugate pairs for real coefficients, and build real polynomials from complex roots.
  13. 13Building polynomial models and finding numerical rootsConstruct polynomial functions from conditions, fit vertical scale, and approximate roots that lack convenient exact forms.
Source record

References used for this unit

  • Lippman and Rasmussen, Precalculus Volume 1
  • Stitz and Zeager, Precalculus
  • Redden, Advanced Algebra

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.