To multiply a single term over a bracket, you multiply the term outside of the bracket by everything inside of the bracket. This process is often referred to as "expanding the bracket". After expanding, you can often simplify the expression by collecting like terms.
Understanding the Process
The core concept revolves around the distributive property. This property states that a(b + c) = ab + ac. In simpler terms, the term 'a' is distributed (multiplied) across both 'b' and 'c' inside the parentheses.
Steps to Multiply a Single Term Over a Bracket
Here's a step-by-step guide:
- Identify the term outside the bracket: This is the term that will be multiplied by each term inside.
- Identify the terms inside the bracket: List out each individual term inside the bracket, paying close attention to the signs (+ or -) in front of them.
- Multiply: Multiply the term outside the bracket by each term inside the bracket.
- Simplify (if possible): After expanding, combine any like terms to simplify the expression. Like terms have the same variable raised to the same power (e.g., 3x and 5x are like terms, but 3x and 5x² are not).
Examples
Let's illustrate this with some examples:
Example 1:
3(x + 2)
- Term outside the bracket: 3
- Terms inside the bracket: x, +2
- Multiply: 3 x + 3 2 = 3x + 6
- Simplified expression: 3x + 6
Example 2:
-2(2y - 5)
- Term outside the bracket: -2
- Terms inside the bracket: 2y, -5
- Multiply: -2 2y + (-2) (-5) = -4y + 10
- Simplified expression: -4y + 10
Example 3 (including simplification):
4(a + 2b) - a
- Expand the bracket: 4 a + 4 2b - a = 4a + 8b - a
- Simplify by collecting like terms (4a and -a): 3a + 8b
- Simplified expression: 3a + 8b
Common Mistakes to Avoid
- Forgetting to multiply by every term: Ensure you multiply the outer term by each term inside the bracket.
- Incorrectly applying the sign: Pay close attention to the signs (+ or -) when multiplying. A negative multiplied by a negative results in a positive.
- Mixing up like terms: You can only combine terms that have the same variable raised to the same power.
Practical Insights
Understanding how to expand brackets is a fundamental skill in algebra. It's crucial for solving equations, simplifying expressions, and various other mathematical operations. Practice with a variety of examples to become proficient in this technique.
