%5E5.jpeg "(2x-3y)^5")
Expanding (2x - 3y)^5 using the Binomial Theorem
The Binomial Theorem provides a powerful way to expand expressions of the form (a + b)^n. In our case, we want to expand (2x - 3y)^5.
The Binomial Theorem Formula
The Binomial Theorem states:
(a + b)^n = ∑_(k=0)^n (n choose k) * a^(n-k) * b^k
where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
- ∑_(k=0)^n indicates the sum from k = 0 to k = n.
Applying the Formula
Let's apply the Binomial Theorem to our expression:
(2x - 3y)^5 = ∑_(k=0)^5 (5 choose k) * (2x)^(5-k) * (-3y)^k
Now we expand each term:
- k = 0: (5 choose 0) * (2x)^5 * (-3y)^0 = 1 * 32x^5 * 1 = 32x^5
- k = 1: (5 choose 1) * (2x)^4 * (-3y)^1 = 5 * 16x^4 * (-3y) = -240x^4y
- k = 2: (5 choose 2) * (2x)^3 * (-3y)^2 = 10 * 8x^3 * 9y^2 = 720x^3y^2
- k = 3: (5 choose 3) * (2x)^2 * (-3y)^3 = 10 * 4x^2 * (-27y^3) = -1080x^2y^3
- k = 4: (5 choose 4) * (2x)^1 * (-3y)^4 = 5 * 2x * 81y^4 = 810xy^4
- k = 5: (5 choose 5) * (2x)^0 * (-3y)^5 = 1 * 1 * (-243y^5) = -243y^5
Final Result
Finally, combining all the terms, we get the expanded form of (2x - 3y)^5:
(2x - 3y)^5 = 32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 - 243y^5