%5E5.jpeg "(x^4-y)^5")
Exploring the Expansion of (x^4 - y)^5
The expression (x^4 - y)^5 represents a binomial raised to the power of 5. Expanding this expression can be achieved using the Binomial Theorem. This theorem provides a systematic way to determine the coefficients and terms in the expansion.
The Binomial Theorem
The Binomial Theorem states:
(x + y)^n = Σ (n choose k) * x^(n-k) * y^k
Where:
- n is the power to which the binomial is raised
- k is an integer ranging from 0 to n
- (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!)
Expanding (x^4 - y)^5
Let's apply the Binomial Theorem to expand (x^4 - y)^5:
-
Identify n and y: In this case, n = 5 and y = -y.
-
Calculate the binomial coefficients: We need to calculate (5 choose k) for k = 0, 1, 2, 3, 4, and 5. These values are:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
-
Apply the theorem: Substitute the values into the Binomial Theorem:
(x^4 - y)^5 = (5 choose 0) * (x^4)^5 * (-y)^0 + (5 choose 1) * (x^4)^4 * (-y)^1 + (5 choose 2) * (x^4)^3 * (-y)^2 + (5 choose 3) * (x^4)^2 * (-y)^3 + (5 choose 4) * (x^4)^1 * (-y)^4 + (5 choose 5) * (x^4)^0 * (-y)^5
-
Simplify the terms:
(x^4 - y)^5 = x^20 - 5x^16y + 10x^12y^2 - 10x^8y^3 + 5x^4y^4 - y^5
Conclusion
The expanded form of (x^4 - y)^5 is x^20 - 5x^16y + 10x^12y^2 - 10x^8y^3 + 5x^4y^4 - y^5. The Binomial Theorem provides a powerful tool for expanding expressions of this form. Understanding the theorem and its application can be useful in various mathematical and scientific contexts.