The binomial Theorem allows us to expand many brackets without multiplying each bracket out one by one. It states: To expandwe could expandwhich would be a very long winded process. Or we could just substitute forandinto the expression for the binomial expansion. Example: Expandthen which simplifies to and further to We may be asked to […]

## Matrices and Practical Problems

Data can usefully be summarised in a table, and a table can have it’s borders, column and row labels taken away and then enclosed in brackets in which case it be come a matrix. Then we can perform useful calculations with it. For instance: The figures in the table show The daily production, in kilograms, […]

## Using Matrices to Solve Simultaneous Equations

If you already know how to solve simultaneous equations then you may well wonder why people use matrices to solve them. The fact is, while simple equations with two unknowns x and y are quite easy to solve, as the number of unknowns increases so does the number of equations we have to solve. The […]

## Practical Vectors

Planes rarely fly in the direction they are pointing. If the wind is blowing and the world is turning the pilot has to take account of these when he plots a course. Even with modern gps systems available, , it is beneficial to the pilot to take these into account because of the resulting increase […]

## Optimisation – Maximising a Function

As a race we need the most from our limited resources. Every manager must decide what or who to send where so as to get the job done with the least effort and risk, and at the least cost, or decide what products to make and sell, subject to the available labour and machine time, […]

## Curves and Maxima/Minima/Stationary Points/Turning Points

I don’t know why these thing need four possible names. We can call them stationary points and classify them as maxima or minima. Definition: A stationary point is a point on a curve where Definition: A stationary point, withis a Minima if On a graph a minimum is lower than the points on either side […]

## Relative Velocities and Relative Positions

Definition: If A is at positionmoving with velocityand B is at positionmoving with velocitythen The relative position of B relative to A is given byand the relative velocity of B relative to A is given by The relative position of A relative to B is given byand the relative velocity of A relative to B […]

## The Factor and Remainder Theorems

Long division with polynomials sounds, and is, a great deal more complicated than long division with numbers. Fortunately though, it is not always necessary. There are two very helpful theorems which often turn the problem of long division into one of substitution. The Factor Theorem: Ifis a factor ofthensois also a root ofor equivalently, a […]

## Symbols and Notation

– the set of positive integers and zero – x is greater than y is greater than – x is less than y x >=y – x is greater than or equal to y – the set of integers, – the set of positive integers, – the set of rational numbers – the set of […]

## Solving Simple Differential Equations

Simple differential equations take the formWe have to solve the equation to findas a function ofWe do this by putting all the ‘s on the right and integrating. Normally when we integrate we have to add a constant. We can find the value of this constant if we are told a point on the curve. […]