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Simplifying (x + y)³
The expression (x + y)³ represents the cube of the binomial (x + y). To simplify it, we can use the distributive property and the concept of binomial expansion. Here's how:
Expanding (x + y)³
- Rewrite the expression: (x + y)³ = (x + y)(x + y)(x + y)
- Expand the first two factors: (x + y)(x + y) = x(x + y) + y(x + y) = x² + xy + xy + y² = x² + 2xy + y²
- Multiply the result by (x + y): (x² + 2xy + y²)(x + y) = x²(x + y) + 2xy(x + y) + y²(x + y)
- Distribute: x³ + x²y + 2x²y + 2xy² + xy² + y³
- Combine like terms: x³ + 3x²y + 3xy² + y³
The Formula
You can also use the following formula to directly expand (x + y)³:
(x + y)³ = x³ + 3x²y + 3xy² + y³
This formula is a specific case of the binomial theorem, which provides a general way to expand expressions of the form (x + y)ⁿ.
Example
Let's simplify (2a + 3b)³ using the formula:
(2a + 3b)³ = (2a)³ + 3(2a)²(3b) + 3(2a)(3b)² + (3b)³ = 8a³ + 36a²b + 54ab² + 27b³
Therefore, the simplified form of (2a + 3b)³ is 8a³ + 36a²b + 54ab² + 27b³.
Conclusion
Simplifying (x + y)³ involves expanding the expression using the distributive property or applying the binomial theorem formula. This process helps you express the expression in a more concise and expanded form, making it easier to understand and manipulate in various mathematical contexts.