Expanding multiple brackets involves systematically multiplying each term in one bracket by each term in the other bracket(s). The approach varies slightly depending on the number of brackets you're dealing with. Here's a comprehensive guide:
Expanding Two Brackets
Method
- Distribute the first term: Multiply the first term of the first bracket by each term in the second bracket.
- Distribute the second term: Multiply the second term of the first bracket by each term in the second bracket.
- Combine like terms: Add up any terms with the same variable and exponent.
Example
Let's expand (a + b)(c + d):
- First term distribution: a c + a d = ac + ad
- Second term distribution: b c + b d = bc + bd
- Combine: ac + ad + bc + bd
Expanding Three Brackets
Method
According to the reference, when expanding triple brackets, we multiply the first two brackets together, then multiply the result with the final bracket.
- Expand the first two: Apply the two-bracket expansion method to the first two brackets.
- Multiply by the third: Multiply the result of the first step by each term in the third bracket.
- Combine like terms: Simplify by combining like terms.
Example
Let's expand (a + b)(c + d)(e + f):
- Expand the first two: From the previous example, we know that (a + b)(c + d) = ac + ad + bc + bd
- Multiply by the third: Now, we multiply (ac + ad + bc + bd) by (e + f):
- (ac + ad + bc + bd) * e = ace + ade + bce + bde
- (ac + ad + bc + bd) * f = acf + adf + bcf + bdf
- Combine: Combining these results we get ace + ade + bce + bde + acf + adf + bcf + bdf.
General Approach for More Brackets
- The principle remains the same. You would multiply the first two brackets together and then multiply the result by the next bracket.
- Repeat this process until all brackets are multiplied.
- Always ensure you distribute each term correctly and combine like terms for simplification.
Practical Insights
- FOIL Method: For two binomial brackets (brackets with two terms) like (a + b)(c + d), you can remember the acronym FOIL - First, Outer, Inner, Last. This helps recall all term distributions.
- Care with Negatives: Be careful with negative signs. Distributing -a is like multiplying each term in the bracket by -1 and then by a.
- Organization: When dealing with many terms, stay organized by writing each multiplication clearly to avoid errors.
- Simplification: Always combine like terms after each multiplication step to keep the process manageable.
- Order: The order of multiplication doesn't affect the final answer, but starting with the first two makes the process straightforward.
Summary Table
| Number of Brackets | Method |
|---|---|
| Two Brackets | Distribute each term of the first bracket across all terms in the second bracket, then combine like terms. |
| Three Brackets | Multiply the first two brackets, then multiply the result with the third bracket, then combine like terms. |
| More than Three | Repeat the process of multiplying two brackets at a time until all brackets are accounted for. |
