Units 1 & 2 (Foundation of Polynomials)
Focus on understanding polynomial expressions, operations (addition, subtraction, multiplication, division), factorization techniques, the Factor and Remainder Theorems, and the graphical behavior and transformations of low-degree polynomial functions.

• Unit 2 (Advanced Algebra & Prelude to Calculus)
  Extend to composite and inverse functions, solving polynomial equations (including systems), and introduce techniques that bridge into calculus (e.g. rates of change via tangent approximations on polynomial graphs).

• Units 3 & 4 (Calculus Applications)
  Differentiate and integrate polynomial functions by rule, analyze turning points, inflection points, and area under curves, and apply these in modelling contexts
  
  Unit 1: Foundations of Polynomial Algebra and Graphs

  1.1 Introduction to Polynomials

  • Definition & Terminology: monomial, binomial, trinomial; degree; leading coefficient.

  • Polynomial Arithmetic

    • Addition/subtraction of like terms

    • Multiplication: distributive law, FOIL, expanding higher-degree products.

  1.2 Factorization Techniques

  • Common Factor Extraction

  • Quadratic Factorization: splitting the middle term, completing the square

  • Higher Degree: grouping, synthetic division.

  • Remainder and Factor Theorems:

    • Using f(a)f(a)f(a) to test factors

    • Polynomial long division vs. synthetic division

  1.3 Graphing Polynomial Functions

  • Key Features: intercepts, turning points (at most n−1n-1n−1 for degree nnn), end-behavior based on leading term.

  • Sketching Practice: simple cubics and quartics.

  • Technology Integration: TI-Nspire/Casio graphing verification.

  1.4 Transformations of the Plane

  • Horizontal & Vertical Shifts

  • Reflections & Stretches: impact on polynomial shape

  • Composite Transformations

  • Modelling Contexts: e.g. height of a projectile as a quadratic function.

  Unit 2: Extending Polynomial Concepts & Bridging to Calculus

  2.1 Composite and Inverse Polynomial Functions

  • Composite: (f∘g)(x)(f\circ g)(x)(f∘g)(x), domain considerations

  • Inverse: when a polynomial has an inverse, solving f(x)=yf(x)=yf(x)=y for xxx.

  2.2 Solving Polynomial Equations

  • Exact Solutions via factorisation

  • Graphical & Numerical Approaches: where exact solutions are not required (e.g. approximate roots via technology)

  • Systems Involving Polynomials: e.g. solving simultaneously with linear functions.

  2.3 Introduction to Rates of Change

  • Average Rate over an interval on a polynomial curve

  • Secant vs. Tangent Line: geometric interpretation

  • Numerical Estimation of Instantaneous Rate as a prelude to differentiation.