# Algebraic Expressions

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## Algebraic Expressions

**There are a number of different ways of writing algebraic expressions, such as: **

1. | (x + 1)^{2} |
An Expression |

2. | (x + 1)^{2} ≡ x^{2} + 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.

**For example:**

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} ≡ x^{2} + 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.)*

**Expanding Brackets**

There is only one rule - multiply everything in the bracket by the expression outside.

**For example:**

2x(3x − 1) = 6x^{2} − 2x

**For example:**

(3x - 2)(4x - 5) = 3x(4x - 5) - 2(4x - 5) = 12x^{2} - 15x - 8x + 10 = 12x^{2} - 23x +10

**Factorising**

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**.

**For example:**

4x^{2} + 12x = 4x (x + 3)

When factorising a **quadratic** expression (ax^{2} + bx + c), you will need **two brackets**.

**To factorise a quadratic:**

- Look for values that can multiply to make the x
^{2}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:**

4x^{2} + 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.

**For example:**

5x^{3} − 30x^{2} + 25x = 5x (x^{2} − 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.