Algebra and Complex Equations

• Covers: This module provides an in-depth exploration of solving various types of equations, distinguishing between real and complex roots. It begins by covering techniques for manipulating quadratics and polynomials, introducing the Factor and Remainder Theorems, and solving quadratic, surd, and power equations with real roots, along with examining the properties of polynomial roots. The module then introduces complex numbers, explaining powers of 'i', manipulation of complex numbers, modulus and argument, complex conjugates, and converting between rectangular and polar forms, including multiplication, division, and using De Moivre's Theorem for powers. Finally, it focuses on solving equations with complex roots, including quadratics, polynomials, and power equations, and introduces the concept of the locus of a point.

• Outcomes:  Manipulate and solve quadratic, polynomial, surd, and power equations to find real roots.  Apply the Factor and Remainder Theorems effectively.  Understand and perform operations with complex numbers in both rectangular and polar forms.  Calculate modulus, argument, and complex conjugates, and apply De Moivre's Theorem.  Solve quadratic, polynomial, and power equations that yield complex roots.  Comprehend and determine the locus of a point in the complex plane.