%5E7.jpeg "(x-2y)^7")
Expanding (x - 2y)^7 using the Binomial Theorem
The Binomial Theorem provides a systematic way to expand expressions of the form (x + y)^n. In this case, we'll be expanding (x - 2y)^7.
The Binomial Theorem Formula
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)!). This signifies the number of ways to choose k items from a set of n items.
- ∑_(k=0)^n represents the sum from k = 0 to k = n.
Expanding (x - 2y)^7
Let's apply the Binomial Theorem to our expression:
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Identify n: In this case, n = 7.
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Expand the sum: We'll need to calculate the terms for k = 0, 1, 2, 3, 4, 5, 6, and 7.
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Calculate the binomial coefficients: For each value of k, we'll calculate (7 choose k).
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Substitute x and y: In our expression, x = x and y = -2y.
Let's break down the expansion term by term:
k = 0:
- (7 choose 0) = 1
- x^(7-0) = x^7
- (-2y)^0 = 1
- Term: x^7
k = 1:
- (7 choose 1) = 7
- x^(7-1) = x^6
- (-2y)^1 = -2y
- Term: -14x^6y
k = 2:
- (7 choose 2) = 21
- x^(7-2) = x^5
- (-2y)^2 = 4y^2
- Term: 84x^5y^2
k = 3:
- (7 choose 3) = 35
- x^(7-3) = x^4
- (-2y)^3 = -8y^3
- Term: -280x^4y^3
k = 4:
- (7 choose 4) = 35
- x^(7-4) = x^3
- (-2y)^4 = 16y^4
- Term: 560x^3y^4
k = 5:
- (7 choose 5) = 21
- x^(7-5) = x^2
- (-2y)^5 = -32y^5
- Term: -672x^2y^5
k = 6:
- (7 choose 6) = 7
- x^(7-6) = x
- (-2y)^6 = 64y^6
- Term: 448xy^6
k = 7:
- (7 choose 7) = 1
- x^(7-7) = 1
- (-2y)^7 = -128y^7
- Term: -128y^7
Final Expanded Form
Combining all the terms, the complete expansion of (x - 2y)^7 is:
(x - 2y)^7 = x^7 - 14x^6y + 84x^5y^2 - 280x^4y^3 + 560x^3y^4 - 672x^2y^5 + 448xy^6 - 128y^7