This is Chapter 2 – Real Polynomials for the Year 12 Specialist Mathematics course.

This chapter extends students’ earlier work with quadratics, cubics and quartics into a more general and structured study of polynomial functions. Rather than focusing only on solving individual equations, students examine the properties that apply to polynomials of any degree and learn how roots, factors, coefficients and graphs are connected.

We begin by revisiting operations with polynomials and developing fluency with notation and algebraic manipulation. Zeros, roots and factors are then linked formally, allowing students to move between an equation, its factorised form and the intercepts of its graph.

Polynomial equality requires students to recognise when two expressions are identical for every value of the variable and to compare corresponding coefficients. Polynomial division provides a systematic method for reducing higher-degree expressions and leads naturally into the Remainder and Factor Theorems. These theorems allow students to evaluate and factorise polynomials efficiently without always carrying out full division.

The Fundamental Theorem of Algebra establishes that a polynomial of degree has exactly complex roots when multiplicity is counted. This creates an important bridge to the later study of complex numbers. Students also investigate relationships between coefficients and the sum and product of roots.

Graphing cubic and quartic polynomials brings the algebraic ideas together. Students consider intercepts, multiplicities, end behaviour and repeated roots to predict how a graph crosses or touches the axis.

Polynomial reasoning supports later work in complex numbers and calculus and appears throughout mathematical modelling, engineering, computer graphics and numerical analysis.

Subchapters:

• 2A: Operations with polynomials

• 2B: Zeros, roots, and factors
• 2C: Polynomial equality
• 2D: Polynomial division
• 2E: The remainder theorem
• 2F: The factor theorem
• 2G: The fundamental theorem of algebra
• 2H: Sum and product of roots theorem
• 2I: Graphing real polynomials - cubics
• 2J: Graphing real polynomials - quartics

By the end of this chapter, students will be able to analyse, divide, factorise and graph higher-degree polynomials while explaining the relationships between roots, factors and coefficients.