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SingaporeFurther Maths

Complex Numbers and Polynomials

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we represent complex numbers in Cartesian, polar and exponential form, and what does the Argand diagram show?

Represent complex numbers in Cartesian, polar and exponential form, perform arithmetic, and interpret them on the Argand diagram
A focused answer to the H2 Further Mathematics outcome on complex numbers. Cartesian, polar (modulus-argument) and exponential forms, conjugates, arithmetic, the geometry of the Argand diagram, and converting between forms.
11 min answer →

How does de Moivre's theorem let us take powers of complex numbers and derive trigonometric identities?

State and apply de Moivre's theorem to find powers of complex numbers and to derive multiple-angle and power-reduction trigonometric identities
A focused answer to the H2 Further Mathematics outcome on de Moivre's theorem. The statement for integer powers, using it to expand multiple angles, deriving cos and sin of n-theta, and the z plus one over z method for power-reduction identities.
11 min answer →

How do equations and inequalities involving the modulus and argument define loci and regions on the Argand diagram?

Sketch loci and regions in the Argand diagram defined by conditions on the modulus and argument of a complex number
A focused answer to the H2 Further Mathematics outcome on loci in the complex plane. Circles from modulus conditions, perpendicular bisectors from equal-distance conditions, half-lines from argument conditions, and shading regions defined by inequalities.
11 min answer →

How do the roots of a polynomial relate to its coefficients, and how do complex roots occur in conjugate pairs?

Use the relationships between the roots and coefficients of a polynomial and apply the conjugate root theorem for real polynomials
A focused answer to the H2 Further Mathematics outcome on polynomials. The sum and product of roots, symmetric functions of roots, the conjugate root theorem for real polynomials, and forming new equations whose roots are transformed.
11 min answer →

What are the nth roots of unity and of a general complex number, and how are they arranged geometrically?

Find the nth roots of unity and the nth roots of a general complex number, and describe their geometric arrangement on the Argand diagram
A focused answer to the H2 Further Mathematics outcome on roots of unity. The n nth roots of unity, their arrangement as a regular polygon, finding the nth roots of a general complex number, and the sum of the roots.
11 min answer →