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*

Lessons

#### Using Partial Fractions

- Video - Partial Fractions (4:24)
- Topic - Equating Polynomials 1
- Topic - Equating Polynomials 2
- Topic - Equating Polynomials 3
- Topic - Factorised Denominator 1
- Topic - Factorised Denominator 2
- Topic - Factorised Denominator 3
- Topic - Perfect Square Denominator 1
- Topic - Perfect Square Denominator 2
- Topic - Perfect Square Denominator 3
- Topic - Degree of Numerator = Degree of Denominator
- Topic - Degree of Numerator > Degree of Denominator

#### Complex Conjugates

- Video - Conjugate Pairs of Complex Numbers (3:54)
- Topic - Writing Conjugate Pairs of Complex Numbers
- Topic - Writing Conjugate Pairs of Complex Numbers using Notation
- Topic - Proof using Conjugate Pairs 1
- Topic - Proof using Conjugate Pairs 2
- Topic - Quadratic Equations with Real Coefficients Having Complex Roots 1
- Topic - Quadratic Equations with Real Coefficients Having Complex Roots 2

#### Conjugates and Division of Complex Numbers

- Video - Realising the Denominators using Conjugate Pairs (6:28)
- Video - Proof regarding Conjugate Pairs (4:13)
- Video - Proof regarding Conjugate Pairs & Realising Denominators (2:20)
- Topic - Division of Complex Numbers 1
- Topic - Division of Complex Numbers 2
- Topic - Multiplicative Inverse of Complex Numbers 1
- Topic - Multiplicative Inverse of Complex Numbers 2

#### Square Roots of Complex Numbers

- Video - Square Roots of Complex Numbers in Positive Real Parts (13:45)
- Video - Square Roots of Complex Numbers in Negative Real Parts (4:50)
- Video - Square Roots of Complex Numbers using Factorise Method 1 (7:03)
- Video - Square Roots of Complex Numbers using Factorise Method 2 (5:33)
- Video - Square Roots of Complex Numbers with Irrational Coefficients (2:43)

#### Modulus

- Topic - Modulus of Complex Numbers
- Topic - Modulus of Purely Imaginary Complex Numbers
- Topic - Modulus of Purely Real Complex Numbers
- Topic - Modulus of Conjugate Pairs
- Topic - Modulus of Polar Form
- Topic - Proof Regarding Modulus
- Topic - Proof Regarding Modulus and Conjugate Pairs
- Topic - Modulus Squared and Multiple of Complex Number and its Conjugate
- Topic - Finding Modulus by Polar Form

#### Complex Factors of Polynomials

- Topic - Factorise two-term quadratics over complex number field 1
- Topic - Factorise two-term quadratics over complex number field 2
- Topic - Factorise three-term quadratics over complex number field 1
- Topic - Factorise three-term quadratics over complex number field 2
- Topic - Factorise three-term quadratics over complex number field 3
- Topic - Factorise two-term quartics over complex number field
- Topic - Factorise three-term quartics over complex number field
- Topic - Find all factors of cubic polynomials over complex number field 1
- Topic - Find all factors of cubic polynomials over complex number field 2
- Topic - Find all factors of cubic polynomials over complex number field 3
- Topic - Find all factors of quartic polynomials over complex number field
- Topic - Complex roots and conjugate root theorem 1
- Topic - Complex roots and conjugate root theorem 2
- Topic - Complete Polynomials with Imaginary Coefficients using Factors 1
- Topic - Complete Polynomials with Imaginary Coefficients using Factors 2

#### Equations over Complex Number Field

- Video - Complex Roots of Quadratic Equations (4:06)
- Video - Complex Roots of Cubic Equations (2:35)
- Video - Complex Roots of Quadratic Equations using Quadratic Formula (4:19)
- Video - Quadratics in Complex Number Field using Quadratic Formula (2:41)
- Video - Quadratics in Complex Number Field using Completing Square (7:54)

#### Modulus-Argument (Polar) Form

- Video - Understanding Modulus-Argument Form (8:13)
- Topic - Horizontal Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Vertical Complex Numbers to Modulus-Argument (Polar) Form
- Topic - First Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Second Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Third Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Fourth Quadrant Complex Numbers to Modulus-Argument (Polar) Form

#### Conversion between Cartesian and Polar Forms

- Video - Complex Numbers in 1st Quarter to Mod-Arg Form (4:26)
- Video - Complex Numbers in 2nd Quarter to Mod-Arg Form (2:33)
- Video - Complex Numbers in 4th Quarter to Mod-Arg Form (4:29)
- Video - Complex Numbers in Purely Real to Mod-Arg Form (2:23)
- Video - Complex Numbers in Purely Imaginary to Mod-Arg Form (1:47)
- Video - Modulus-Argument Form to Cartesian Form (5:38)
- Video - Mod-Arg Form to Cartesian Form in Pure Real or Imaginary (4:23)
- Video - Simplifying Complex Numbers using Mod-Arg Form (3:45)
- Video - Exact Values of Sine & Cosine using Mod-Arg Form (4:57)

#### De Moivre's Theorem

- Video - Understanding De Moivre's Theorem (2:44)
- Video - Proof of De Moivre's Theorem Multiplication (3:42)
- Video - Proof of De Moivre's Theorem Division (2:20)
- Video - Proof of De Moivre's Theorem Subtraction (4:04)
- Video - Simplifying Complex Numbers using De Moivre's Theorem (7:42)
- Video - Purely Imaginary using De Moivre's Theorem (5:11)
- Video - Purely Real using De Moivre's Theorem (5:24)
- Video - Trigonometric Expression using De Moivre's Theorem (5:27)
- Video - De Moivre's Theorem involving Conjugate Pairs (6:58)
- Video - Trigonometric Properties & De Moivre's Theorem (6:04)

#### Roots of Unity in Complex Numbers

- Video - Nth Roots of Unity using Complex Numbers (8:04)
- Video - Complex Roots of Unity (6:03)
- Video - 4th Roots of Unity using Complex Numbers (3:09)
- Video - 5th Roots of Unity using Complex Numbers (1:57)
- Video - 6th Roots of Unity using Complex Numbers (2:30)
- Video - Cube Roots of Negative Unity using Complex Numbers (2:09)
- Video - Seventh Root of Unity (6:46)

#### Roots of Complex Numbers

- Video - 4th Roots of 4 using Complex Numbers (2:32)
- Video - 5th Roots of -32 using Complex Numbers (2:45)
- Video - Cube Roots of i using Complex Numbers (2:19)
- Video - 4th Roots of -16i using Complex Numbers (1:42)
- Video - 5th Roots of 1+i using Complex Numbers (2:07)
- Video - Cube Roots of Irrational Real Part & i using Complex Numbers (1:30)
- Video - Exact Values of Trigonometric Ratios involving Fifth Roots (6:37)

#### Complex Roots of Polynomials

- Video - Complex Roots of Real Coefficients in Polynomials (3:26)
- Video - Writing Polynomial Equations of Smallest Degree with Imaginary Roots (4:21)
- Video - Quadratic Factors given Complex Roots of Quartic Polynomials (2:22)
- Video - Writing Rational Factors of Quartic Polynomials (3:40)
- Video - Writing Real Factors of Quartic Polynomials (1:00)
- Video - Writing Complex Factors of Quartic Polynomials (0:52)
- Video - Long Division of Cubic Polynomials by Complex Factors (7:08)