Expanding Parentheses. How to Multiply out a Simple Bracket.

In math you will often have to multiply out parentheses (brackets). The correct name given to multiplying out a parenthesis is call expanding. The term at the start of the bracket must multiply the values inside the bracket. Let’s take a look at a few examples of multiplying out parentheses.

Example 1

Expand 3(2x + 4y)

The 3 at the start of the parenthesis must multiply both terms inside the bracket:

3 × 2x = 6x

3 × 4y = 12y

Putting these terms together gives a final expression of 6x + 12y.

Example 2

Expand 7(3x - 2y)

The 3 at the start of the parenthesis must multiply both terms inside the bracket:

7 × 3x = 21x

7 × -2y = -14y

Putting these terms together gives a final expression of 21x – 14y

Example 3

Expand -5(4k – 2z)

The -5 at the start of the parenthesis must multiply both terms inside the bracket:

-5 × 4k = -20k

-5 × -2z = 10z

On this example you will need to take care with the -5. Remember if you multiply a negative by a positive number you get a negative answer, and if you multiply a negative by a negative you get a positive answer.

Putting these terms together gives a final expression of -20k + 10z

Example 4

Expand x(x+4)

The x at the start of the parenthesis must multiply both terms inside the bracket:

x × x = x²

Make sure you remember that x times x is x squared

x × 4 = 4x

Putting these terms together gives a final expression of x² + 4x

Example 5

Expand 7x(9x + 4a)

The 7x at the start of the parenthesis must multiply both terms inside the bracket:

7x × 9x = 63x²

7x × 4a = 28ax

Make sure you put the letters in each term in alphabetical order.

Putting these terms together gives a final expression of 63x² + 28ax

Example 6

Expand 6(2x + 3y – 2z)

The 6 at the start of the parenthesis must multiply the three terms inside the bracket:

6 × 2x = 12x

6 × 3y = 18y

6 × -2z = -12z

Putting these terms together gives a final expression of 12x + 18y – 12z.

The same method can be applied to multiplying out two single brackets. First off all multiply out the first bracket, and then multiply out the second bracket. Be carful if there is a negative number between the two brackets. Once the two brackets are expanded, the like terms can be simplified by adding or subtracing them. Lets take a look at an example of expanding two single brackets.

Example 7

Expand and simplify:

3(2x + 4y) + 2(x +y)

First multiply out the first single bracket to give 6x + 12y.

Next multiply out the second bracket to give 2x + 2y.

Now putting this altogether you have 6x + 12y + 2x + 2y.

Now simplify the expression to give by adding the x terms together and the y terms together to give 8x + 14y.

Example 8

Expand and simplify:

5(2x - 3y) - 2(x - 4y)

First multiply out the first single bracket to give 10x - 15y.

Next multiply out the second bracket to give -2x + 8y.

Now putting this altogether you have 10x - 15y - 2x + 8y.

Now simplify the expression by combining the x terms and y terms to give 8x - 7y.

So to summarise, if you are asked to multiply out a bracket, the term at the start of the bracket must multiply the terms inside the bracket. Make sure you put the letters of each term in alphabetical order with the number first. Also if you have to multiply a x by a x then this will be x squared. Also if you have two pairs of single brackets take care multiplying out the second bracket if there is a negative number at the start of the second bracket.

Once you have mastered single brackets you can then move onto expanding double brackets. Double brackets can be multiplied out a similar way, most people use the FOIL method, which involves multiplying out the first terms, the outer terms the inner terms and the last terms. So for example, you may be asked to expand and simplify (x+3)(x+2). The first terms will give x^2, the outer terms will give 2x, the inner terms will give 3x and the outer terms will give 6. Putting this altogether and simplifying the middle two terms gives a final answer of x^2 + 5x + 6.