Typical surd expressions take the form To simplify an expression like this we multiply numerator (top) and denominator (bottom) by the conjugate root of the bottom. In this case the conjugate root isThe ontain an expression of the formover d whereare whole numbers in all these equations. Example: Rationalise the denominator and simplify: Example: Rationalise […]

## Using Discriminants to Find the Number of Roots of a Quadratic Curve

In general a quadratic curve may have one, two or no roots, as shown. For the curve the number of roots depends entirely on the discriminant Ifthere are no roots. Ifthere is one root. Ifthere are two roots. We are typically asked: Find the set of values offor which the curve has no roots: always, […]

## Integration

We have to integrate functions of the form . The rule here is to add one to the power and divide by the new power, never forgetting that we must add a constant. This extends to any combination of sums and multiples of powers of For example, remembering that which may be simplified further. This […]

## Determining the Equation of a Quadratic From a Graph

Every quadratic graph is unique, since if two quadratic graphs pass through the same set of points, they must be the same graph. The general equation of a quadratic graph is(1). This has three constantsandso we can use three points on the graph to write down three simultaneous equations and solve them to findand Example: […]

## Sigma Notation

The sigma symbolsignifies summation. means add up all the numbersThere are several different, similar variations of notation, so thatandmean essentially the same thing, though the second example means explicitly ‘add up all thefromto‘. Example: Iffind Example: Iffind Summation in this way has several useful properties: and Summation is also linear This last property can be […]

## Long Division of Polynomials

Long division of one number by another, if the divisor is not a factor, results in a decimal number, or a quotient plus a remainder. For exampleremainder 1 or3 is the quotient and 1 the remainder. It is not just pure numbers that can undergo long division. So can polynomials. In the example below it […]

## Curve Sketching

When sketching curves of a polynomial function try to factorise it first, if it is not already factorised. Writing a polynomial as a product of factors –– for example – makes it easier to identify the roots – the points whereThe roots will be points on the x – axis, since each root is a […]

## Integrating and Differentiating Algebraic Fractions

Integration of any algebraic fraction is quite simple if the denominator is a power ofWe can just simplify the numerator to express it as a sum of powers ofdivide each term by the denominator so the whole expression is a sum of powers ofthen integrate each term individually by adding one to the power and […]

## Lines of Symmetry for Quadratic Curves

Every quadratic curve has a line of symmetry – it is a basic property of quadratic curves. For example the curveshown below has a line of symmetry about the line For a more complicated quadratic, for example,we have to complete the square. In this case we get The line of symmetry is given […]

## Circular Measure

Here we are concerned with lengths of arcs and areas of segment and sectors. The circumference of a circle is 2%pi r. If we haven’t got a whole circle or we want to find the length of just part of it’s circumference, we find find the cicumference of just that part of the circle. […]