Expanding double brackets means multiplying everything inside the first bracket by everything inside the second bracket.
What Double Brackets Mean
When you see two brackets written next to each other, it signifies multiplication. As stated in the reference, writing two brackets next to each other means the brackets need to be multiplied together. For example, (y + 2)(y + 3) means (y + 2) × (y + 3). The goal of expanding is to remove the brackets and express the result as a single, simplified algebraic expression.
Methods for Expanding
The underlying principle is the distributive property: each term in the first bracket must multiply each term in the second bracket.
For expanding two binomials (expressions with two terms, like (a + b) and (c + d)), a common and memorable method is FOIL.
The FOIL Method
FOIL is an acronym that helps you remember which pairs of terms to multiply:
- First: Multiply the first term in each bracket.
- Outer: Multiply the outer terms (the first term in the first bracket and the second term in the second bracket).
- Inner: Multiply the inner terms (the second term in the first bracket and the first term in the second bracket).
- Last: Multiply the last term in each bracket.
After multiplying these four pairs, you combine any like terms to simplify the expression.
Step-by-Step Example using FOIL
Let's expand the example from the reference: (y + 2)(y + 3).
Here are the steps using the FOIL method:
- First: Multiply the first term in
(y + 2)by the first term in(y + 3).
y * y = y² - Outer: Multiply the outer term in
(y + 2)(y) by the outer term in(y + 3)(3).
y * 3 = 3y - Inner: Multiply the inner term in
(y + 2)(2) by the inner term in(y + 3)(y).
2 * y = 2y - Last: Multiply the last term in
(y + 2)(2) by the last term in(y + 3)(3).
2 * 3 = 6
Now, collect all the results:
y² + 3y + 2y + 6
Finally, combine any like terms (the 3y and 2y):
3y + 2y = 5y
So, the expanded and simplified expression is:
y² + 5y + 6
Summary of the Example Expansion
Here's a table illustrating the FOIL steps for (y + 2)(y + 3):
| Step | Terms to Multiply | Calculation | Result |
|---|---|---|---|
| First | y and y |
y * y |
y² |
| Outer | y and 3 |
y * 3 |
3y |
| Inner | 2 and y |
2 * y |
2y |
| Last | 2 and 3 |
2 * 3 |
6 |
Combined results before simplification: y² + 3y + 2y + 6
After combining like terms (3y + 2y): y² + 5y + 6
Key Takeaways
- Double brackets side-by-side mean multiplication.
- You must multiply every term in the first bracket by every term in the second bracket.
- Methods like FOIL provide a structured way to do this for binomials.
- Always simplify by combining like terms after multiplying.
Following these steps allows you to successfully expand double brackets and simplify algebraic expressions.
