(x+y)^5 Expand

Jasonbradley

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Jun 17, 2024

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Expanding (x + y)^5: A Step-by-Step Guide

Expanding expressions like (x + y)^5 can seem daunting, but there's a straightforward method using the Binomial Theorem. Let's break down the process step by step.

Understanding the Binomial Theorem

The Binomial Theorem provides a formula for expanding any binomial raised to a power:

(x + y)^n = ∑ (n choose k) * x^(n-k) * y^k

Where:

• n is the power the binomial is raised to.
• k is a counter ranging from 0 to n.
• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Expanding (x + y)^5

1. Identify n: In our case, n = 5.

2. Calculate binomial coefficients:

   • (5 choose 0) = 5! / (0! * 5!) = 1
   • (5 choose 1) = 5! / (1! * 4!) = 5
   • (5 choose 2) = 5! / (2! * 3!) = 10
   • (5 choose 3) = 5! / (3! * 2!) = 10
   • (5 choose 4) = 5! / (4! * 1!) = 5
   • (5 choose 5) = 5! / (5! * 0!) = 1

3. Apply the formula:

   (x + y)^5 = (5 choose 0) * x^5 * y^0 + (5 choose 1) * x^4 * y^1 + (5 choose 2) * x^3 * y^2 + (5 choose 3) * x^2 * y^3 + (5 choose 4) * x^1 * y^4 + (5 choose 5) * x^0 * y^5

4. Simplify:

   (x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

Key Observations

• Coefficients: Notice how the coefficients of the expanded terms follow a symmetrical pattern (1, 5, 10, 10, 5, 1). This pattern is reflected in Pascal's Triangle.
• Exponents: The exponents of x decrease from n to 0, while the exponents of y increase from 0 to n.
• Symmetry: The expanded terms are symmetrical. The first term is x^5, the last is y^5, and the middle terms mirror each other.

Alternative Methods

While the Binomial Theorem is a powerful tool, you can also expand (x + y)^5 by repeated multiplication:

1. (x + y)^2 = (x + y)(x + y) = x^2 + 2xy + y^2
2. (x + y)^3 = (x + y)^2 (x + y) = (x^2 + 2xy + y^2)(x + y) = x^3 + 3x^2y + 3xy^2 + y^3
3. (x + y)^4 = (x + y)^3 (x + y) = (x^3 + 3x^2y + 3xy^2 + y^3)(x + y) = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
4. (x + y)^5 = (x + y)^4 (x + y) = (x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4)(x + y) = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

Conclusion

Understanding the Binomial Theorem allows you to efficiently expand expressions like (x + y)^5 and gain valuable insights into the pattern of coefficients and exponents. This knowledge is essential for various mathematical applications, including calculus, probability, and statistics.