1. Fundamental Mathematics

1.1

Sum and product of roots of quadratic equations.

1.2

The factor and remainder theorems and their application to the solutions of polynomial equations.

1.3

Partial fractions for rational functions including repeated linear and quadratic factors.

See The Method of Partial Fractions, from S.O.S. MATHematics.

Also see The Method of Partial Fractions, from the Review of Algebra Techniques, part of Oregon State University's extensive Web Study Guide project.

The University of Saskatchewan's Exercises in Math Readiness (EMR) site has a section on Partial Fraction Decompositions. Included are an introduction and three sets of exercises : Introductory, Moderate, and Advanced. Recommended.

1.4

Inequalities in one variable, including their graphical representation.
The manipulation of inequalities, including the use of the modulus (absolute value) sign.

See Solving Absolute Value Inequalities, from the University of Wisconsin Marathon Center's Notes for College Algebra and Geometry.

Also see Inequalities and Graphical Methods for Solving Inequalities, both from the University of Saskatchewan's Exercises in Math Readiness (EMR) site.

1.5

Solution of radical equations, including equations with extraneous solutions.

For an introduction to radicals, see Square Roots and Other Radicals, from Exercises in Math Readiness (EMR).

1.6

Solution of pairs of equations, linear or non-linear, in two variables.

1.7

Sequences and series, including arithmetic and geometric series.
Sum to infinity of geometric series where appropriate.

See The On-Line Encyclopedia of Integer Sequences. Among other topics, this interesting site discusses "puzzle" sequences, "classic" sequences, and "hot" sequences. Recommended.

1.8

Proof by mathematical induction, standard results and other applications.
Forming conjectures to be proved by mathematical induction.

A proof of the principle of mathematical induction may be found at Countable Infinity.
Explanations are given at Mathematical Induction and Mathematical Induction, two different sites with the same name.

1.9

Counting principles, including permutations and combinations.

The (Combinatorial) Object Server covers a wide number of subjects, including Information on Permutations and Information on Combinations of a Set.

The Combinatorics site provided by Oberlin College has chapters (with exercises) on
The Multiplication Principal, Permutations and Combinations. Recommended.

For challenging questions, see the problem sets in Combinatorics Topics for K-8 Teachers.

Also see Thoughts on teaching Permutations, Combinations and the Binomial theorem.

1.10

The binomial theorem for a positive integer exponent.
Proof and applications.

See Arithmetic Properties of Binomial Coeffiecients.

Also see Expanding Binomials, from Exercises in Math Readiness (EMR).

1.11

Solution of triangles using the sine and cosine rules.

1.12

Derivation and use of the given trigonometric identities.
The derivation of further identities.

1.13

Solutions of simple trigonometric equations, including methods of substitution.

Simple trigonometric equations are dealt with at Trigonometric Equations, from the University of Texas El Paso's S.O.S. MATHematics site.

1.14

The use of the compound formula   a cos q + b sin q = R cos(q ± a).

1.15

Complex numbers. The number . The terms complex number, real part and imaginary part, conjugate, modulus and argument. The complex plane (Argand diagram).

See The Complex Plane, from S.O.S. MATHematics.

1.16

The form  z  =   a + bi  =  r(cos q + i sin q) and the use of   z   =   re^iq (Euler's formula).

See The Polar Form of a Complex Number : The unit circle, from S.O.S. MATHematics.
De Moivre's theorem is mentioned.

The same site has an explanation of   e^iq = cos q + i sin q. See Euler's Formula.

1.17

Sums, products and quotients of complex numbers.


2 + i is multiplied by i, shown in red, resulting in a rotation of p/2 in the complex plane.
The diagram was created using downloadable packages developed by METRIC :
Mathematics Education Technology Research at Imperial College.
Their learning modules, based on Mathematica, have resulted in a text :
Experiments in Undergraduate Mathematics.

1.18

De Moivre's Theorem. Powers and roots of a complex number.

For background material, see The Polar Form of a Complex Number : The unit circle.
Proof of the theorem   (cos q  +  i   sin q )ⁿ   =   cos n q + i sin n q   is left as an exercise.

The 12 roots of 1.
Points representing the roots are symmetrically arrayed about the origin.
Graphed using Mathematica and METRIC packages.

1.19

Conjugate roots of polynomial equations with real coefficients.

Polynomial equations are discussed at Polynomials and Roots, from the University of Saskatchewan's Exercises in Math Readiness (EMR). The Advanced Exercises require the solution of equations having radicals as coefficients.

1.20

Roots of simple equations with complex coefficients.

1.21

The concept of a locus in the Cartesian and complex planes.
Geometrical and algebraic methods of determining loci.


The locus of   z < |0.7|   in the complex plane.
Graphed using Mathematica and METRIC packages.