(x-y)^4 Binomial Expansion

4 min read Jun 17, 2024
(x-y)^4 Binomial Expansion

Expanding the Binomial Power: (x-y)^4

The binomial theorem provides a systematic way to expand expressions of the form (x + y)^n, where n is a non-negative integer. Let's explore how to apply this theorem to expand the specific case of (x - y)^4.

Understanding the Binomial Theorem

The binomial theorem states that for any real numbers x and y, and any non-negative integer n:

(x + y)^n = Σ (n choose k) x^(n-k) y^k

where the summation is from k = 0 to n, and (n choose k) represents the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Applying the Binomial Theorem to (x-y)^4

Let's break down the expansion step-by-step:

  1. Identify n: In our case, n = 4.
  2. Expand the summation: We need to calculate the terms for k = 0, 1, 2, 3, and 4.
  3. Calculate the binomial coefficients:
    • (4 choose 0) = 4! / (0! * 4!) = 1
    • (4 choose 1) = 4! / (1! * 3!) = 4
    • (4 choose 2) = 4! / (2! * 2!) = 6
    • (4 choose 3) = 4! / (3! * 1!) = 4
    • (4 choose 4) = 4! / (4! * 0!) = 1
  4. Substitute the values into the formula:

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

  1. Simplify:

**(x - y)^4 = ** x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4

Key Observations

  • Alternating Signs: Notice that the terms in the expansion alternate between positive and negative signs. This is because we are expanding (x - y), introducing a negative sign with each increasing power of y.
  • Coefficients: The binomial coefficients (1, 4, 6, 4, 1) follow a symmetrical pattern, forming what is known as Pascal's Triangle.

Conclusion

By applying the binomial theorem, we have successfully expanded (x - y)^4 into a polynomial expression with five terms. The resulting equation provides a valuable tool for understanding the relationship between x and y when raised to a specific power. This expansion can be utilized in various mathematical contexts, including algebra, calculus, and statistics.

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