There are a number of different ways of writing algebraic expressions, such as:
|1.||(x + 1)2||An Expression|
|2.||(x + 1)2 ≡ x2 + 2x + 1||An Identity|
|3.||(x + 1)2 = 3x + 6||An Equation|
|4.||4x - 5 < x + 1||An Inequality|
(x + 1)2 is an expression (there is no '=' or 'inequality' sign) that can have different values depending on the value we give to x.
If we give x the value of 2, then
(x + 1)2 = (2 + 1)2 = 9
This expression can also be referred to as a function of x, where we use f as the symbol for function.
This would give us:
f(x) = (x + 1)2
If we were working with x = 2, we would write,
f(2) = (2+1)2 = 9
(x + 1)2 ≡ x2 + 2x + 1 is called an identity because it is true for all values of x. (This means both sides of the equation are identical.) Note that in an identity the '=' sign is replaced by the '≡' sign (to say that it is always true for every value of x).
Much of A level Mathematics is spent rewriting expressions in a different way - (all those 'Show that' questions) - and each of these rewrites will be a new identity for the original expression.
There are several basic skills we must have in order to succeed in A level Mathematics. In particular we must be able to expand brackets, and factorise.
(These topics were also essential GCSE skills. If you find them hard then practice them again and again until you can do them quickly and easily.)
There is only one rule - multiply everything in the bracket by the expression outside.
2x(3x − 1) = 6x2 − 2x
(3x - 2)(4x - 5) = 3x(4x - 5) - 2(4x - 5) = 12x2 - 15x - 8x + 10 = 12x2 - 23x +10
This is the inverse (opposite) of expanding brackets and involves rewriting expressions with either 1 or 2 brackets.
Use one bracket when there is a common factor.
4x2 + 12x = 4x (x + 3)
When factorising a quadratic expression (ax2 + bx + c), you will need two brackets.
To factorise a quadratic:
- Look for values that can multiply to make the x2 term.
- Look for values that multiply to make the constant, c.
- Choose a combination that results in the correct multiple of x.
Before you continue, try factorising this quadratic:
4x2 + x − 5
The answer you get should be: (4x + 5)(x − 1)
Sometimes you may need a combination of both skills to factorise more complex expressions.
5x3 − 30x2 + 25x = 5x (x2 − 6x + 5) = 5x(x − 1)(x − 5)
Completing the Square, the Binomial Expansion, Partial Fractions (see later), and Trigonometric Identities (see the Trigonometry topics), are other examples of working with identities.