VCE Specialist Mathematics Units 3 and 4 Algebra
Course

VCE Specialist Mathematics Units 3 and 4 – Algebra

2.1 Rational Functions
2.2 Complex Numbers

28 Lessons

In this area of study students cover the expression of simple rational functions as a sum of partial fractions; the arithmetic and algebra of complex numbers, including polar form; points and curves in the complex plane; introduction to factorisation of polynomial functions over the complex field; and an informal treatment of the fundamental theorem of algebra.

This area of study includes:

Rational functions of a real variable, including:

  • definition of a rational function and expression of rational functions of low degree as sums of partial fractions.

Complex numbers, including:

  • \(C\), the set of numbers \(z\) of the form \(z = x + yi\) where \(x, y\) are real numbers and \( i^2 = –1\), real and imaginary parts, complex conjugates, modulus
  • use of an argand diagram to represent points, lines, rays and circles in the complex plane
  • equality, addition, subtraction, multiplication and division of complex numbers
  • polar form (modulus and argument); multiplication and division in polar form, including their geometric representation and interpretation, proof of basic identities involving modulus and argument
  • De Moivre’s theorem, proof for integral powers, powers and roots of complex numbers in polar form, and their geometric representation and interpretation
  • \(n^{th}\) roots of unity and other complex numbers and their location in the complex plane
  • factors over \(C\) of polynomials with integer coefficients; and informal introduction to the fundamental theorem of algebra
  • factorisation of polynomial functions of a single variable over \(C\), for example, \(z^8+1, z^2 –i, z^3–(2–i)z^2 + z–2 + i\)
  • solution over \(C\) of corresponding polynomial equations by completing the square, factorisation and the conjugate root theorem.

source – VCE Mathematics Study Design

RATIONAL FUNCTIONS

Rational Functions

Further Rational Functions

Using Partial Fractions

COMPLEX NUMBERS

Negative Discriminant and Imaginary Numbers

Introduction to Complex Numbers

Operations with Complex Numbers

Equality of Complex Numbers

Complex Conjugates

Conjugates and Division of Complex Numbers

Properties of Conjugates

Square Roots of Complex Numbers

2-D Vectors

Modulus

Complex Number Plane or Argand Diagram

Complex Factors of Polynomials

Equations over Complex Number Field

Connection to Coordinate Geometry

Modulus-Argument (Polar) Form

Properties of Argument

Conversion between Cartesian and Polar Forms

De Moivre's Theorem

Roots of Unity in Complex Numbers

Roots of Complex Numbers

Factorising using Complex Numbers

Complex Roots of Polynomials

Complex Polynomials