# VCE Specialist Mathematics Units 3 and 4 – Algebra

2.1 Rational Functions

2.2 Complex Numbers

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*

VCE Specialist Mathematics Units 3 and 4 Courses

VCE Specialist Mathematics Units 3 and 4 Syllabus

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### Rational Functions

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Complex Numbers

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Complex Conjugates
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2-D Vectors
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Modulus
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