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## Multiplying out brackets

### Multiplying a bracket by a single term

• We must take account of the sign of the term
• Each and every term in the bracket is multiplied by the term out side

### Examples

$3(x+5)=3x+15$

In the example above we multiplied the x and 5 by 3

This is illustrated in the diagram where one side is of length 3 and the other is $x+5$.  The area is the required answer.

$x(x-4)=x^2-4x$

In the example above we multiplied the x and -4 by x

$-2(y-3)=-2y+6$

In the example above we multiplied the y and -3 by -2

$4+2(x-5)=4+2x-10=-6+2x$

In the example above the +2 is the only entity that is multiplying the contents of the bracket

### Multiplying two brackets together

Consider the problem $(x+3)(x+6)$

If we look at the illustration above we can see that each element of the first bracket is multiplied by each element of the second bracket.

The result is that $(x+3)(x+6)=x^2+6x+3x+18=x^2+9x+18$

To help make sure that we multiply every thing in the first bracket by everything in the second bracket (provided that there are exactly two in both) we can use the mnemonic FOIL

1. F multiply the first element in each bracket together
2. O multiply the outside elements of the expression together
3. I multiply the inside elements of the expression together
4. L multiply the last element in each bracket together

### Examples

$(x+2)(x-5)=x^2-5x+2x-10=x^2-3x-10$

Take notice of the order that I reached my terms (I used FOIL) and the fact that I simplified the expression afterwards.

$(2t-2)(3t-5)=6t^2-10t-6t+10=6t^2-16t+10$

Notice that multiply by the directed (signed) value not just the positive value

### Multiplying multiple different sized brackets together

• When you multiply brackets together make sure that all elements the first bracket are multiplied by all elements of the second
• Multiply a pair of brackets together first, if there are more than two, to reduce the number of brackets by one
• Keep on doing this until all brackets have be multiplied out

Example

$(2x+3)(x-1)(x+2)$

multiply the first two brackets together

$=(2x^2-2x+3x-3)(x+2)$

Simplify the new first bracket

$=(2x^2+x-3)(x+2)$

Multiply out the brackets.  Because there are three terms in the first bracket and two in the second we would expect to get $2 \times 3=6$ terms in the expansion

$=2x^3+4x^2+x^2+2x-3x-6$

$=2x^3+5x^2-x-6$