%5E6.jpeg "(x-y)^6")
The Expansion of (x-y)^6
The expansion of (x-y)^6 can be determined using the Binomial Theorem. This theorem provides a general formula for expanding expressions of the form (x + y)^n, where n is a positive integer.
The Binomial Theorem
The Binomial Theorem states:
(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)!). This represents the number of ways to choose k objects from a set of n objects.
Expanding (x-y)^6
Applying the Binomial Theorem to (x - y)^6, we get:
(x - y)^6 = ∑_(k=0)^6 (6 choose k) x^(6-k) (-y)^k
Let's expand this:
- k = 0: (6 choose 0) x^6 (-y)^0 = x^6
- k = 1: (6 choose 1) x^5 (-y)^1 = -6x^5y
- k = 2: (6 choose 2) x^4 (-y)^2 = 15x^4y^2
- k = 3: (6 choose 3) x^3 (-y)^3 = -20x^3y^3
- k = 4: (6 choose 4) x^2 (-y)^4 = 15x^2y^4
- k = 5: (6 choose 5) x^1 (-y)^5 = -6xy^5
- k = 6: (6 choose 6) x^0 (-y)^6 = y^6
Therefore, the complete expansion of (x - y)^6 is:
(x - y)^6 = x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6
Key Points
- The coefficients in the expansion of (x-y)^6 follow a pattern known as Pascal's Triangle.
- The signs alternate between positive and negative.
- The powers of x decrease from 6 to 0, while the powers of y increase from 0 to 6.
Understanding the Binomial Theorem and its application allows for efficient expansion of expressions like (x-y)^6, simplifying complex calculations.