%5E4-binomial-expansion.jpeg "(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:
- Identify n: In our case, n = 4.
- Expand the summation: We need to calculate the terms for k = 0, 1, 2, 3, and 4.
- 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
- 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
- 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.