## Expanding and Factorising

### Expanding in Numbers

There are two ways to calculate 4(5 + 6)

 Method 1 To calculate 4(5 + 6) This means 4 × (5 + 6) Using the BEDMAS rule the expression in the brackets must be calculated first. 4 × 11 Do the multiplying. 44

OR

 Method 2 To calculate 4(5 + 6) this means 4 × 5 + 4 × 6 Using the BEDMAS rule the multiplying is done before the adding. 20 + 24 Do the adding. 44

This second method is called the distributive property and can also be used in algebra.

### Expanding in Algebra

The expression 2(p + 3) means:

multiplied by (p + 3)

OR

lots of (p + 3)

Therefore, if the brackets are removed then the result will be 2 lots of p and 2 lots of 3

Thus 2(p + 3) = 2 × p + 2 × 3 = 2p + 6

Removing brackets from an expression is known as expanding the expression.

This usually occurs when a term outside a bracket is multiplied by every term inside the bracket.

The outside term "distributes" itself over the inside terms. Thus the name the distributive property.

### Distributive Property in Action

When an expression is expanded, each term inside the bracket is multiplied by the term outside the bracket.

Example 1

 3(p + 4) means 3 multiplied by (p + 4) = 3.p + 3.4 = 3p + 12 When brackets have been expanded, the expression must be simplified further if possible.

Example 2

5(y + 7) = 5 x y + 5 x 7 = 5y + 35

 Further examples Expanded form Expand and simplify: 3(p + 8) 3p + 24 4(2x − 5) 8x − 20 -2(x − 3) -2x + 6 5a(a − 2) 5a2 − 10a

Note When expanding a bracket with a NEGATIVE number in front, the signs of each term inside the bracket change.

### Factorising Factorising an algebraic expression is the opposite process of expanding brackets.
It involves placing brackets into an expression.
Once an expression has been factorised, it should then be expanded to check it is correct.
Some expressions cannot be factorised.

### Common Factors

common factor is a number or letter that divides into each term of an expression.

If every term of the expression has a common factor, this common factor should be taken out and placed in front of a set of brackets.

 Unfactorised Expression Common factor Other factor Factorised expression 2p + 4 2 (p + 2) 2(p + 2) 4x − 8 4 (x − 2) 4(x − 2) 12m + 18n 6 (2m + 3n) 6(2m + 3n) 3a2 + 6a 3a (a + 2) 3a(a + 2)