There is only so much to go around, even more so now with the Earth’s resources running low and the price of everything. We have to make the most of what we have, and decide what things should be used for so that the most use can be got out of them.. W will consider […]

## The Binomial Theorem and Estimation

The binomial expansion can in certain circumstances give highly accurate estimates of certain powers or roots. This can happen for example when we want to find not too high a power of a number that is close to an integer. The binomial expansion is given by Suppose we are asked to findto 5 decimal places. […]

## Solving Quadratic Exponential Equations by Substitution

Some exponential equations can be factorised in linear factors. The simplest can be factorised into quadratic equations. We then put each factor equal to zero and solve it. Example: Solve(1) Factorise to get or The above equation has two solutions. In general, as for quadratic equations, an exponential which can be expressed as two factors […]

## Simplifying Logarithmic Expressions

It will first be relevant to summarise the rules of logarithms: (1) (2) (3) (4) (5) Often we can simplify a logarithmic expression so that it becomes either a single log, or a constant plus a log. Example: Simplify using (1) using (3) using (5) Example: Ifexpressterms of p using (1) using (3) Simplify using […]

## Transformations of Graphs

The graphs given by the equationare called hyperbolae. One is illustrated below. In fact there are a set of equations which all give the same shape of graph as shown above. The graphs may be moved up or down, translated, rotated or any combination of these, but the shapes of the graphs are still fundamentally […]

## The Remainder Theorem

The Remainder Theorem is the generalized form of The Factor Theorem. The Remainder Theorem When a polynomial expressionis divided by a linear factorthe remainder isIfthenis a factor of Example: Show thatis a factor of thereforeis a factor. Example If the remainder whenis divided byis 3 and the remainder whenis divided byis 4, findand Remainder ofon […]

## Geometric Series

A geometric series is such that each term is multiplied by a fixed number to get the next term. 1, 2, 4, 8, 16… is a geometric series because each term is multiplied by a number called the common ratio – in this case 2, to get the next term. We may write We can […]

## 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 […]

## The Difference Quotient – Differentiating From First Principles

The gradient of the straight line between two pints is given byWe may generalize this for a curve to find the gradient at any point, but we need to take into account that the curve is actually a curve. In more basic maths we can draw a tangent at a point and find the gradient […]

## Arithmetic Sequences

An arithmetic sequence is a sequence such that the difference between successive terms is constant. 2, 6, 10, 14, 18 has a constant common difference term 4. We add 4 to each term to get the next term. We can write down the rule: Arithmetic sequences can be defined iteratively, so that each term is […]