%5E3-expanded-form.jpeg "(x+y)^3 Expanded Form")
Expanding (x+y)^3: A Step-by-Step Guide
When working with algebraic expressions, it's essential to understand how to expand expressions with exponents. In this article, we'll focus on expanding (x+y)^3, a cubic expression that can be a bit tricky to handle. But don't worry, we'll break it down step by step, making it easy to grasp.
The Formula
To expand (x+y)^3, we can use the binomial theorem, which states that:
(x+y)^n = x^n + nx^(n-1)y + n(n-1)x^(n-2)y^2 + … + y^n
where n is a positive integer. In our case, n = 3.
Expanding (x+y)^3
Let's apply the binomial theorem to expand (x+y)^3:
(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
Here's how we got each term:
- x^3: This is the first term, where x is raised to the power of 3.
- 3x^2y: This term is obtained by multiplying x^2 (the second power of x) by y, and then multiplying by 3 (the coefficient).
- 3xy^2: This term is obtained by multiplying x (the first power of x) by y^2 (the second power of y), and then multiplying by 3 (the coefficient).
- y^3: This is the last term, where y is raised to the power of 3.
Simplifying the Expression
The expanded form of (x+y)^3 is:
x^3 + 3x^2y + 3xy^2 + y^3
This is the final answer. You can verify this by plugging in some values for x and y and checking that the expression holds true.
Conclusion
Expanding (x+y)^3 might seem daunting at first, but by using the binomial theorem, we can break it down into manageable parts. Remember to follow the formula and apply it step by step to get the correct answer. With practice, you'll become proficient in expanding algebraic expressions and tackling more complex problems.
