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Topic 1: Further complex numbers

Complex arithmetic using polar form

3 hours

• Prove complex number identities involving modulus and argument, e.g. z z̅ = |z|², |z₁||z₂| = |z₁z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).
• Use De Moivre’s theorem for integral powers. 1 interactive

Roots of complex numbers

3 hours

• Determine and examine the nth roots of unity and their location on the unit circle. 1 interactive
• Determine and examine the nth roots of complex numbers and their location in the complex plane. 1 interactive

Factorisation of polynomials

5 hours

• Apply the factor theorem and the remainder theorem for polynomials.
• Understand and use the complex conjugate root theorem for polynomials with real coefficients, e.g. factorise a cubic polynomial with real coefficients given one factor. 1 interactive
• Solve polynomial equations over ℂ to order 4 including those with real and imaginary coefficients, e.g. solve z⁴ + 3z³ − 2z² + iz − 2 = 0 and z³ − 2iz² + z − 2i = 0.