Polynomials

Area of Study- Functions and Graphs (unit 1)

From the VCAA 2016-2020 Mathematics Study Design

In this area of study students cover the graphical representation of simple algebraic functions (polynomial and power functions) of a single real variable and the key features of functions and their graphs such as axis intercepts, domain (including the concept of maximal, natural or implied domain), co-domain and range, stationary points, asymptotic behaviour and symmetry. The behaviour of functions and their graphs is explored in a variety of modelling contexts and theoretical investigations.

Area of Study- Algebra (unit 1)

This area of study supports students’ work in the ‘Functions and graphs’, ‘Calculus’ and ‘Probability and statistics’ areas of study, and content is to be distributed between Units 1 and 2. In Unit 1 the focus is on the algebra of polynomial functions of low degree and transformations of the plane.

Key skills and knowledge (outlined from the VCAA Study Design for Outcome 1)

• the key features and properties of power and polynomial functions and their graphs
• the effect of transformations of the plane, dilation, reflection in axes, translation and simple combinations of these transformations, on the graphs of linear and power functions
• factorisation patterns, the quadratic formula and discriminant, the remainder, factor and rational root theorems and the null factor law
• express a cubic polynomial p (x), with integer coefficients, in the form p (x) = (x – a) q (x) + r and determine p (x) / (x-a), by hand
• use algebraic, graphical and numerical approaches, including the factor theorem and the bisection method, to determine and verify solutions to equations over a specified interval
• apply distributive and index (exponent) laws to manipulate and simplify expressions involving polynomial and power function, by hand in simple cases
• sketch by hand graphs of linear, quadratic and cubic polynomial functions, and quartic polynomial functions in factored form (approximate location of stationary points only for cubic and quartic functions), including cases where an x-axis intercept is a touch point or a stationary point of inflection
• draw graphs of polynomial functions of low degree, simple power functions and simple relations that are not functions

Assessment Overview

Topic tests will assess:

• key knowledge and skills,
• ability to problem-solve and use the mathematics in context,
• use of the CAS, to assist problem-solving