Expanding single brackets is required as a stand-alone algebraic skill but it is also required for rearranging formulae and solving equations. In this article we look at what is meant by expanding single brackets and how to do it.
Expanding Single Brackets
What Does ‘Expanding Single Brackets’ Mean?
Expanding single brackets means to multiply the term outside of the bracket by every term inside of the bracket. Expanding a single bracket is the opposite of factorising a single bracket and can be useful in solving equations.
In the above example, to expand 5(-3x + 7) we have to multiply 5 by every term inside the brackets.
So we multiply 5 by -3x to get -15x. Then we multiply 5 by +7 to get +35.
Hence, our answer is -15x + 35.
How to Expand Single Brackets
In order to expand single brackets, you have to multiply every term inside the bracket by the term outside of the bracket. In most questions there will be two terms inside the bracket but there can be three or more terms to multiply.
- Multiply the first term inside the bracket by the term outside of the bracket.
- Multiply the second term inside the bracket by the term outside of the bracket.
- Make sure you have the correct plus or minus sign between the two terms in your answer.
- If you have more than two terms inside the bracket, multiply these terms by the term outside of the bracket, making sure you have the correct plus or minus sign between them.
Table Method Examples
We can use a table method to multiply every term inside the bracket.
Example 1: Two Terms
Example 2: Two Terms Including a Negative
Note: Don’t forget to include the sign in front of the term.
Example 3: Two Negative Coefficients
If you have a negative sign outside of the bracket don’t forget to multiply the negative into the bracket. In this example we are multiplying by -5 (not 5).
Example 4: Variables with Coefficients Greater Than 1
Multiply both the coefficients and the variables. Remember that x × x = x².
Example 5: Three Terms (Including Variables with Coefficients Greater Than 1)
‘Multiplying Out’ Method
Once you become more confident using the table method you may be able to multiply mentally without the need for a table.
The use of arrows can be helpful in demonstrating which terms are being multiplied. The following examples may help to demonstrate.
Example 1: Multiplying Out Single Brackets
If doing this without the use of a table make sure to double check the positive and negatives signs of each term as it is a common mistake to write the wrong sign.
Example 2: Multiplying Out Single Brackets (3 terms)
Expanding Single Brackets Problems
Area Problem
To visualise expanding a single bracket we can imagine a rectangle whose area is calculated by multiplying the length by the width.
We can calculate the total area by adding the two smaller areas (as in method 1).
Alternatively we can find the total area by multiplying the width by the total length of both rectangles, which is 3 × (x + 7). We write this as 3(x + 7) which then needs to be expanded.
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