Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.310NA
recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems
Content
 Students:
 Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems (ACMNA266)
 recognise a polynomial expression \( a_n x^n + a_{n1} x^{n1} + \ldots + a_1 x + a_0 \) and use the terms 'degree', 'leading term', 'leading coefficient', 'constant term' and 'monic polynomial' appropriately
 use the notation \(P(x)\) for polynomials and \(P(c)\) to indicate the value of \(P(x)\) for \(x=c\)
 add and subtract polynomials and multiply polynomials by linear expressions
 divide polynomials by linear expressions to find the quotient and remainder, expressing the polynomial as the product of the linear expression and another polynomial plus a remainder, ie \( P(x) = (xa)Q(x) + c \)
 verify the remainder theorem and use it to find factors of polynomials
 use the factor theorem to factorise particular polynomials completely
 use the factor theorem and long division to find all zeros of a simple polynomial \(P(x)\) and then solve \(P(x)=0\) (degree ≤ 4)
 state the number of zeros that a polynomial of degree \(n\) can have
 Apply an understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation (ACMNA268)
 sketch the graphs of quadratic, cubic and quartic polynomials by factorising and finding the zeros
 recognise linear, quadratic and cubic expressions as examples of polynomials and relate the sketching of these curves to factorising polynomials and finding the zeros (Reasoning)
 use digital technologies to graph polynomials of odd and even degree and investigate the relationship between the number of zeros and the degree of the polynomial (Communicating, Problem Solving)
 connect the roots of the equation \(P(x)=0\) to the \(x\)intercepts, and the constant term to the \(y\)intercept, of the graph of \(y=P(x)\) (Communicating, Reasoning)
 determine the importance of the sign of the leading term of a polynomial on the behaviour of the curve as \( x \to \pm \infty \) (Reasoning)
 determine the effect of single, double and triple roots of a polynomial equation \(P(x)=0\) on the shape of the graph of \(y=P(x)\)
 use the leading term, the roots of the equation \(P(x) = 0\), and the x and yintercepts to sketch the graph of \(y=P(x)\)
 describe the key features of a polynomial and draw its graph from the description (Communicating)
 use the graph of \(y=P(x)\) to sketch \(y=P(x)\), \(y=P(x)\), \(y=P(x)+c\), \(y=kP(x)\)
 explain the similarities and differences between the graphs of two polynomials, such as \(y=x^3+x^2+x\) and \(y=x^3+x^2+x+1\) (Communicating, Reasoning)
Purpose/Relevance of Substrand
Polynomials are important in the further study of mathematics and science. They have a range of realworld applications in areas such as physics, engineering, business and economics. Polynomials are used in industries that model different situations, such as in the stock market to predict how prices will vary over time, in marketing to predict how raising the price of a good will affect its sales, and in economics to perform cost analyses. They are used in physics in relation to the motion of projectiles and different types of energy, including electricity. As polynomials are used to describe curves of various types, they are used in the real world to graph and explore the use of curves, eg a roadbuilding company could use polynomials to describe curves in its roads.