This document introduces the algebraic exercises on this site. They are divided into different categories and include trigonometric problems in addition to pure algebraic ones. The grading of difficulty is from high-school third form to university entrance levels. Each exercise sets out to give the student practice in a particular kind of algebraic manipulation - from the basic substitution of a known value into an expression up to the solution of polynomial equations, trigonometric functions and the use of series and difference equations.

The example sheets, which each contain 20 individual exercises, are provided as PDF files for printing so that they may be used alongside the worked out answers by students. This has been deliberately chosen since it is frequently the choice of method and the detail working of that method which is more important didactically than the 'right' answer!

While it is not intended to limit student solutions in any way, it is suggested that they should be encouraged not to use calculators when doing algebraic examples unless real number solutions are required - and then only as late as possible to avoid the effects of ill-conditioned equations.

Substitution

Substitution of a value for an algebraic variable is the simplest form of algebra, since it builds simply on arithmetic skills which have been learnt earlier. It involves evaluating simple algebraic expressions by substituting a given value for the one or more unknown variables to give a numeric result. The examples here range from simple single variable substitution to complex multi-variate polynomial substitution.

Later examples involve partial substitution and simplification only to prepare for the solution of simultaneous equations in the set of simultaneous eqation exercises given later.

1. Single variable, linear - answers here.

Simultaneous Equations

Simultaneous equations of all sorts are given here from the simple two variable pair to the use of up to four variables which it may be more appropriate to solve using matrix methods.

1. Two variable, whole numbers - answers here.
2. Two variable, rational numbers - answers here.
3. Two variable, rational numbers, possibly ill-conditioned - answers here.

Multiplication

Preparatory to tackling the factorisation of polynomials, it is essential to be able to multiply expressions in order to understand the relationships of the individual components to the final resulting polynomial. The exercises in this group are designed to help the student gain insight into the similarities and differences between ostensibly similar factor products.

1. Two linear factors - answers here.
2. Simplifying products

Factorisation

Now that polynomials have been encountered, the student must be able to factorise those with which he is presented when solving some real world problem. The examples here range from the binomial expression to the polynomial in increasing order of difficulty.

1. Binomial expressions - answers here.
2. More Binomial expressions - answers here.

Quadratic Equations

Since the ability to solve quadratic equations lies at the heart of many practical problems, there is an extensive range of exercises from those involving whole number roots to those having complex roots.

• Whole number roots - answers here.
• One rational root - answers here.
• Two rational roots - answers here.
• Two rational roots - answers here.

Trigonometry

Most forms of trigonometric problems involve determining values of functions, while some involve derivation of proofs or solutions to simultaneous, quadratic or other forms of algebraic manipulation.

The examples and exercises here are purely algebraic evaluations and offer no real world context. Where such real world problems are desired, then the problem section should be explored.

• Simple triangles

Further categories of algebraic exercises are being developed and will appear here over time.