%5E5-binomial-expansion.jpeg "(x-y)^5 Binomial Expansion")
Understanding the Binomial Expansion of (x-y)^5
The binomial theorem allows us to expand expressions of the form (x+y)^n, where n is a positive integer. In this case, we'll explore the expansion of (x-y)^5.
The Binomial Theorem
The binomial theorem states that:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). It represents the number of ways to choose k items from a set of n items.
- ∑_(k=0)^n represents the sum from k=0 to n.
Expanding (x-y)^5
Applying the binomial theorem to (x-y)^5, we get:
(x-y)^5 = ∑_(k=0)^5 (5 choose k) * x^(5-k) * (-y)^k
Let's expand this sum term by term:
- k=0: (5 choose 0) * x^(5-0) * (-y)^0 = 1 * x^5 * 1 = x^5
- k=1: (5 choose 1) * x^(5-1) * (-y)^1 = 5 * x^4 * (-y) = -5x^4y
- k=2: (5 choose 2) * x^(5-2) * (-y)^2 = 10 * x^3 * y^2 = 10x^3y^2
- k=3: (5 choose 3) * x^(5-3) * (-y)^3 = 10 * x^2 * (-y^3) = -10x^2y^3
- k=4: (5 choose 4) * x^(5-4) * (-y)^4 = 5 * x * y^4 = 5xy^4
- k=5: (5 choose 5) * x^(5-5) * (-y)^5 = 1 * 1 * (-y^5) = -y^5
Therefore, the complete expansion of (x-y)^5 is:
(x-y)^5 = x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5
Key Points
- The binomial expansion of (x-y)^5 has six terms.
- The coefficients of each term are the binomial coefficients, which can be calculated using the formula (n choose k).
- The signs alternate between positive and negative, starting with a positive sign.
- The powers of x decrease from 5 to 0, while the powers of y increase from 0 to 5.
Understanding the binomial theorem and its application to expanding expressions like (x-y)^5 is crucial in various fields, including algebra, calculus, and probability.