This is Chapter 4 – Polynomials for the Year 11 Mathematical Methods course.

Polynomials extend many of the ideas introduced through quadratics and allow students to explore functions with a much wider range of shapes and behaviours. This chapter is an important step towards the more advanced algebra and function analysis required later in Mathematical Methods, particularly in calculus and Stage 2 polynomial work.

We begin by building fluency with polynomial notation and operations, including expanding, simplifying and working accurately with expressions of different degrees. Students then explore the relationship between zeros, roots and factors, learning that these are different ways of describing the same underlying structure. This creates an important connection between an algebraic equation and the points where its graph crosses or touches the horizontal axis.

Cubic polynomials require a more strategic approach than quadratics. Students will learn how to identify possible factors, test roots systematically and use known factors to reduce a cubic equation into a form that can be solved more easily. These questions reward organised working, pattern recognition and persistence, particularly when the correct path is not immediately obvious.

The chapter then turns towards graphing cubic and quartic functions. Students will examine intercepts, end behaviour, repeated roots and the ways in which factorised form influences the overall shape of a graph. Rather than relying only on plotting technology, the aim is to understand what the algebra reveals before the graph is drawn.

Polynomial functions are used in mathematical modelling, engineering, computer graphics and scientific simulations. More immediately, they provide essential preparation for later calculus work, where students analyse rates of change, turning points and the behaviour of increasingly complex functions.

Subchapters:

• 4A: Polynomials
  
• 4B: Zeros, roots, and factors
• 4C: Factorising cubic polynomials
• 4D: Solving cubic equations
• 4E: Graphing cubic functions
• 4F: Quartic functions

By the end of this chapter, students will be able to manipulate, factorise, solve and graph higher-degree polynomials while clearly connecting their algebraic structure with their graphical behaviour.