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Learning intentions:
In this section we will examine:
- Methods of expansion, including the box method and FOIL
- Collecting like terms
- Patters associated with expansion, including perfect squares and the difference of perfect squares
- Complex expansions involving three or more sets of brackets
Expanding
Expanding brackets involves removing the brackets by multiplying each term in the brackets by every term outside of the brackets. The process of expansion is also known as the distributive law. When expanding brackets be very careful of negative numbers.
Collecting like terms
Once a set of brackets has been expanded, it is possible for several like terms to appear. Recall, like terms contain the same variable (pronumeral) raised to the same power. For example 2x and -7x are like terms.
Expanding when there is only one term outside the brackets
When there is only one term outside the bracket the process of expansion is relatively simple. In this case we multiply each term inside the bracket by the single term outside of the bracket. For example:
Expanding brackets involves removing the brackets by multiplying each term in the brackets by every term outside of the brackets
Like terms contain the same variable raised to the same power.
4A - VIDEO EXAMPLE 1:
Expand the following expression:
4A - VIDEO EXAMPLE 2:
Expand the following expression:
Expanding when there are two brackets present
Expanding becomes more difficult when more terms are outside of the bracket but still need to be multiplied through. When this is the case you can use the box method or FOIL to ensure you expand the set of brackets properly. In general:
Expanding using the box method
To expand using the grid method use the following steps:
- Draw up a 2 x 2 grid/table (or of the appropriate dimension)
- Place the terms inside the first bracket above each of the columns of the grid.
- Place the terms in the second bracket next to each of the rows of the grid.
- Multiply the terms above and next to each cell (box) together and write the product in the cell. It is useful to keep the sign in front even if it is positive (+).
- Add up all of the terms inside the cells of the grid.
- Collect like terms and simplify the expanded expression.
4A - VIDEO EXAMPLE 3:
Expand the following expression using the box method:
4A - VIDEO EXAMPLE 4:
Expand the following expression using the box method:
Expanding using FOIL
To expand using the "FOIL" method use the following steps:
- F - Multiply the first term in each of the brackets together
- O - Multiply the outer terms together; that is, the first term of the first bracket and the last term of the second bracket.
- I - Multiply the two inner terms together; that is, the second term in the first bracket and the first term in the second bracket.
- L - Multiply the last two terms together; that is, the second term in the first bracket and the second term in the second bracket
- Collect like terms and simplify the expanded expression.
Figure 1 - Expanding brackets using the FOIL method.
4A - VIDEO EXAMPLE 5:
Expand the following expression using FOIL:
4A - VIDEO EXAMPLE 6:
Expand the following expression using FOIL:
Patterns associated with expansion
There are two major patterns associated with expansion perfect squares and the difference of perfect squares (DOPS).
Perfect squares
Perfect squares have the following patterns:
- The first and last terms are given by squares of themselves.
- The middle term is given by 2 of the a and b terms multiplied together.
- The first and last terms are given by squares of themselves.
- The middle term is given by -2 of the a and b terms multiplied together.
4A - VIDEO EXAMPLE 7:
Expand the following expression:
4A - VIDEO EXAMPLE 8:
Expand the following expression:
The difference of perfect squares (DOPS)
The difference of perfect squares has the following pattern:
4A - VIDEO EXAMPLE 9:
Expand the following expression:
4A - VIDEO EXAMPLE 10:
Expand the following expression:
More complex expansions
Expanding brackets with more than two terms inside
When brackets contain more than two terms you must be careful to multiply each term inside the bracket by every term outside of the bracket.
4A - VIDEO EXAMPLE 11:
Expand the following expression:
Expanding when more than two sets of brackets are present
When more than two sets of brackets are present:
- Expand two sets of brackets and then place the expanded form in another set of brackets.
- Multiply each of the terms in the expanded form by the terms in the remain bracket.
- Repeat, if necessary, until all brackets have been expanded
4A - VIDEO EXAMPLE 12:
Expand the following expression:
Success criteria:
You will be successful if you can:
- Expand brackets using the box methods and/or FOIL.
- Recognise and apply patterns of expansion, including perfect squares and DOPS.
- Expand more complex expressions involving three or more sets of brackets
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