Advanced Maths: Algebra

By the end of this lesson, you should be able to:

• Understand the definition and structure of rational functions as the ratio of two polynomials.
• Identify whether a given expression is a rational function, even if it contains irrational coefficients.
• Determine the domain and intercepts of rational functions.
• Recognise that rational functions can have asymptotes and complex end behaviour, which may require calculus for full analysis.

By the end of this lesson, you should be able to:

• Understand the idea that certain functions can be expressed as infinite polynomials (Taylor and Maclaurin series).
• Recognise well-known expansions such as those for exponential, sine, and cosine functions.
• Understand the limitations of Taylor series, including convergence conditions and domains of validity.
• Appreciate how polynomial approximations can help in estimating function values.

By the end of this lesson, you should be able to:

• Recognise how the degrees of polynomials in a rational function affect its end behaviour.
• Determine whether a function has a horizontal, diagonal, or no asymptote using limits.
• Use end behaviour analysis to distinguish rational functions from non-rational ones.
• Understand the limitations of this method as a rough check, not a formal proof.

By the end of this lesson, you should be able to:

• Understand the concept of partial fractions and when it can be applied.
• Break down rational functions into simpler fractions using algebraic techniques.
• Use both coefficient comparison and substitution methods to solve for unknown constants.
• Apply partial fraction decomposition to prepare for calculus operations like integration.

By the end of this lesson, you should be able to:

• Recognise when to apply long division before using partial fractions (i.e. when the degree of the numerator is greater than or equal to the denominator).
• Use algebraic long division to simplify a rational function into a polynomial and a proper fraction.
• Decompose the remaining proper fraction into partial fractions using known strategies.
• Apply both long division and partial fractions together in one solution process.

By the end of this lesson, you should be able to:

• Understand the definition of imaginary and complex numbers.
• Identify the real and imaginary parts of a complex number.
• Use notation such as Re(z) and Im(z) to express parts of a complex number.
• Determine whether a number is real, imaginary, or complex.
• Compare two complex numbers and solve for unknowns using equality of real and imaginary parts.

By the end of this lesson, you should be able to:

• Understand what a complex conjugate is and how to find it.
• Identify the modulus of a complex number as its distance from the origin on the Argand diagram.
• Use geometric understanding of complex numbers on the Cartesian (Argand) plane.
• Recognise that complex conjugates reflect across the real axis and that real numbers are their own conjugates.

By the end of this lesson, you should be able to:

• Plot complex numbers on the Argand diagram using the form a + bi.
• Understand that the x-axis represents the real part and the y-axis the imaginary part.
• Visualise complex number operations (addition, subtraction, scalar multiplication) as vector transformations.
• Use vector interpretation to solve and explain results like z₁ + z₂ and z₁ − z₂ geometrically.

By the end of this lesson, you should be able to:

• Recognise that complex numbers follow the same algebraic rules as real numbers.
• Apply key properties such as commutativity, associativity, and distributivity to complex number operations.
• Understand visually how complex addition works using geometric interpretation on the Argand diagram.
• Explain why complex numbers cannot be compared in size using inequality symbols.

By the end of this lesson, you should be able to:

• Add, subtract, and multiply complex numbers by real scalars.
• Recognise that complex number operations behave like vector operations.
• Follow worked examples to understand both algebraic and geometric approaches.
• Interpret complex number operations as geometric transformations on the Argand diagram.
• Connect algebraic calculations with visual movements like translation and scaling on the plane.

By the end of this lesson, you should be able to:

• Multiply two complex numbers using the distributive property (expanding brackets).
• Apply the identity i² = -1 to simplify expressions involving powers of i.
• Use the general formula (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
• Work through examples involving the multiplication of binomial complex expressions.
• Simplify powers of i, such as i², i³, and beyond.

By the end of this lesson, you should be able to:

• Explain how multiplying a complex number by –1 or i affects its position on the Argand diagram.
• Recognise that multiplying by i rotates a complex number 90 degrees anticlockwise.
• Understand the coordinate rule: (a, b) becomes (–b, a) when multiplied by i.
• Identify the repeating pattern in powers of i.

By the end of this lesson, you should be able to:

• Calculate the modulus of a complex number using the formula √(a² + b²).
• Find the conjugate of a complex number by changing the sign of the imaginary part.
• Apply key properties of modulus and conjugate in algebraic operations.
• Understand how conjugates relate to magnitude and real parts in expressions like zz̄ and z + z̄.

By the end of this lesson, you should be able to:

• Divide complex numbers by converting the denominator into a real number using conjugates.
• Use the identity (a + bi)(a − bi) = a² + b² to simplify expressions.
• Find the reciprocal of a complex number using its conjugate.
• Apply these techniques to simplify complex rational expressions.

By the end of this lesson, you should be able to:

• Convert a complex number from Cartesian form (a + bi) to modulus–argument (polar) form.
• Understand and use the form z = r(cos θ + i sin θ), abbreviated as z = r cis θ.
• Identify the modulus r = √(a² + b²) and argument θ from a complex number.
• Interpret and plot complex numbers on a polar coordinate grid.

By the end of this lesson, you should be able to:

• Understand that polar form is not unique because sine and cosine are periodic with period 2π.
• Recognise the standard convention for defining the angle θ within the range –π ≤ θ ≤ π.
• Define and interpret the principal argument of a complex number.

By the end of this lesson, you should be able to:

• Understand that the conjugate of z = r(cos θ + i sin θ) is z̄ = r(cos(–θ) + i sin(–θ)).
• Convert between polar and Cartesian forms using r cos θ and r sin θ.
• Identify the modulus and principal argument of complex numbers through sketching.

By the end of this lesson, you should be able to:

• Understand why polar form is not ideal for addition or subtraction of complex numbers.
• Convert complex numbers from polar to Cartesian form (a + bi) before performing addition or subtraction.
• Carry out addition or subtraction in Cartesian form and convert back to polar form if needed.
• Apply cosine and sine to break down cis notation into real and imaginary components for calculation.

By the end of this lesson, you should be able to:

• Understand how multiplying a complex number by a positive scalar stretches the vector without changing its direction.
• Know that if the scalar is negative, the direction is reversed and the argument changes by ±π.
• Use the form kz = kr cis θ for scalar multiplication in polar coordinates.
• Interpret scalar multiplication geometrically using vector diagrams.

By the end of this lesson, you should be able to:

• Multiply two complex numbers in polar form by multiplying their moduli and adding their arguments.
• Use the formula: z₁z₂ = r₁r₂ cis(θ₁ + θ₂).
• Understand and prove that |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).
• Apply trigonometric identities to support polar multiplication algebraically.

By the end of this lesson, you should be able to:

• Interpret multiplication of complex numbers as scaling and rotating vectors on the Argand diagram.
• Understand that multiplying by z₂ scales the modulus and adds the argument of z₁.
• Recognise that the reciprocal of a complex number has the opposite angle and inverse modulus.
• Use the formula z₁z₂ = r₁r₂ cis(θ₁ + θ₂) to visualise the geometric effect of multiplication.

By the end of this lesson, you should be able to:

• Divide complex numbers in polar form using the formula z₁/z₂ = (r₁/r₂) cis(θ₁ − θ₂).
• Understand that moduli are divided and arguments are subtracted.
• Use properties of reciprocal and conjugate in polar form, such as arg(1/z) = –arg(z).
• Simplify division expressions involving cis notation using angle subtraction and modulus division.

By the end of this lesson, you should be able to:

• Use De Moivre’s Theorem to calculate powers of complex numbers in polar form.
• Understand that raising a complex number to a power scales the modulus and multiplies the angle.
• Interpret the effect of positive and negative powers geometrically as repeated rotations.
• Recognise that multiplying cis(nθ) and cis(–nθ) results in 1, helping to extend the theorem to negative integers.

By the end of this lesson, you should be able to:

• Solve equations of the form zⁿ = a using De Moivre’s Theorem.
• Express complex numbers in polar form and compare moduli and arguments.
• Understand that such equations have n distinct solutions spaced evenly around the unit circle.
• Convert polar solutions to Cartesian form when required.

By the end of this lesson, you should be able to:

• Solve equations of the form zⁿ = a by finding all n complex roots.
• Understand that all solutions lie evenly spaced on a circle centered at the origin.
• Determine the radius and angle of each solution using modulus–argument form.
• Identify that these solutions are called the n-th roots of unity of a.

By the end of this lesson, you should be able to:

• Apply common and advanced polynomial identities
• Simplify and restructure expressions through algebraic manipulation
• Use substitution and expansion to verify identities
• Factorise polynomial expressions efficiently

By the end of this lesson, you should be able to:

• Apply standard and advanced factorisation techniques
• Simplify and reorganise algebraic expressions clearly
• Use substitution and expansion to check correctness of identities
• Factorise polynomial expressions accurately and efficiently

By the end of this lesson, you should be able to:

• Recognise and apply the difference of two squares identity
• Use the sum and difference of cubes identities effectively
• Factorise higher-order polynomials by breaking them into smaller factors
• Apply techniques such as completing the square to obtain complex factors

By the end of this lesson, you should be able to:

• Solve quadratic equations using the quadratic formula, including with complex coefficients
• Apply completing the square method to simplify and solve quadratic expressions
• Use factorisation as an alternative approach when applicable
• Identify special cases such as neat square forms or when the discriminant equals zero

Here’s a concise summary of the lesson on the Conjugate Root Theorem:

• If a + bi is a root of a polynomial with real coefficients, then its conjugate a – bi is also a root.
• The theorem is often applied with factor, remainder, and rational-root theorems to simplify higher-order polynomials.
• Proof for quadratics shows that if α is a root, then its conjugate α̅ must also satisfy the polynomial equation.
• Example: For P(z) = z³ – 3z² + 5z – 3, if z = 1 + √2i is a root, then z = 1 – √2i is also a root, allowing factorisation into linear and quadratic factors.

Here’s a clear summary of the lesson on the Fundamental Theorem of Algebra:

• A polynomial of degree n (where n ≥ 1) always has at least one complex root.
• By repeatedly applying the theorem, any polynomial can be completely factorised into n linear factors over the complex numbers.
• The roots (zeros) of the polynomial may be real or complex, and repeated roots are possible.
• Example to show how rational-root theorem and long division reduce the cubic into linear and quadratic factors, giving solutions.

By the end of this lesson, you should be able to:

• Apply the conjugate root theorem to polynomials with real coefficients
• Use the rational-root theorem to identify possible rational solutions
• Factorise integer-coefficient polynomials using systematic techniques
• Recognise when and how to apply given factors and break polynomials down into simpler parts

By the end of this lesson, you should be able to:

• Recognise and apply techniques for solving equations involving parameters.
• Use root information to reconstruct polynomial equations.
• Match coefficients and compare real and imaginary parts to find unknowns.
• Apply the complex root conjugate rule to ensure real polynomial coefficients.
• Develop confidence with algebraic reasoning as required in VCE Specialist Maths and ATAR-style assessments.

By the end of this lesson, you should be able to:

• Understand the role and structure of algebraic proofs.
• Use proof by contradiction to verify algebraic statements.
• Differentiate between direct proof and proof by contradiction.

By the end of this lesson, you should be able to:

• Interpret equations and inequalities as subsets of the Cartesian plane
• Identify regions formed by intersections of curves and inequalities
• Recognise the differences between Cartesian and Argand planes
• Describe limitations in the Argand plane in terms of vector length and direction

By the end of this lesson, you should be able to:

• Interpret the modulus of a complex number as its distance from the origin
• Represent modulus conditions (e.g., |z| = r, |z| < r) as circles or regions on an Argand diagram
• Translate modulus equations into their equivalent Cartesian equations
• Identify and describe circles with shifted centres in both Argand and Cartesian forms

By the end of this lesson, you should be able to:

• Understand that the argument of a complex number represents its direction from the origin
• Express conditions on the argument (e.g., angle ranges) as regions on an Argand diagram
• Translate argument-based conditions into equivalent Cartesian equations
• Identify and represent vectors with specific directions by shifting the origin in the Argand plane

By the end of this lesson, you should be able to:

• Express the real and imaginary parts of a complex number as Cartesian coordinates
• Set up and interpret relationships between Re(z) and Im(z) in algebraic form
• Translate these relationships into Cartesian equations such as lines, parabolas, and circles
• Represent these conditions visually on an Argand diagram

By the end of this lesson, you should be able to:

• Interpret modulus expressions as distances between points or vectors on an Argand diagram
• Translate distance conditions into Cartesian equations, such as circles or perpendicular bisectors
• Understand how equidistant points from two fixed points form the perpendicular bisector of the segment
• Recognise that more complex distance conditions can lead to standard conic sections, such as ellipses