It is never enough to give lots of examples in maths if you want to prove something. To disprove something you need only give a counterexample, but a proof must be proof positive, conclusive, decisive. For example, suppose you want to prove something simple and intuitive – that the product of two even numbers is […]

## The Second Differential Criterion

If a curve is sloping upis positive. and if a curve is sloping down thenis negative The graph on the left hasincreasing – it goes from negative to zero to positive. This means that the gradient ofis positive. The graph on the right hasdecreasing – it goes from positive to zero to negative. This means […]

## Substitution

Substitution has many uses. It can be used to simplify differential equations and make them easier to solve. It can by use when integrating to transform integrands into forms that can be integrated, it can be used to transform graphs, and used correctly it can make many intractable problems solvable. Example: Find Write down the […]

## Indices and Powers

It is convenient to start with a summary of the rules of indices: Then for example, and We can use these rules in simplifying surds for example: or general products and fractions of powers: or sums of roots: or products of roots: or quotients of roots:

## Solving Quintic Equations

There are general solutions for quintic polynomials – polynomials of order 4. They may be real or not real depending on the polynomial. We are interested here in a special class of quintic polynomials which factorises into two quadratics which we can solve. For example, solve Substituteso thatand the equation becomesThis factorises to givesoor 4, […]

## Factorising Cubic Expressions When One Root is Given

We know how to factorise quadratic equations but it is much harder to factorise cubic because in general there is no easy to see relationship between the coefficients. If we are given a factor or a root however, the problem becomes much more solvable. We can write the cubic expression as a product of a […]

## Completing the Square and Solving Quadratic Equations

Completing the square make it possible to find the maximum or minimum value of a quadratic function without sketching it, or to solve quadratic equations without using the quadratic formula. We start with an expressionto express in the form We might multiply this out to obtainNow we equateto this and solve for the coefficients a, […]

## The Factor Theorem

We can use the factor theorem to find if a particular value ofis a root of a polynomial equation or to find out if a particular linear factor divides a polynomial perfectly, with no remainder. The Factor Theorem Ifis a factor of the polynomialthenhas no remainder or equivalently, We show uses of the factor theorems […]

## The Binomial Theorem

Expanding brackets such as (3+2x)^5 is tedious by expanding the brackets one by one. Fortunately there is a quicker method using the binomial theorem. It states, where n is an integer greater than or equal to one. Example: Expandthen which simplifies to and further to Example: Expandthen This simplifies to We may be asked to […]

## Iterative Sequences

A sequence is a list of numbers with some rule to calculate each number in the sequence. The rule could be a formula for the nth term, for exampleor a formula which says multiply the last term by 2 and add three to get the next term,A sequence generated by a rule which uses each […]