%5E7-binomial-expansion.jpeg "(x-y)^7 Binomial Expansion")
Binomial Expansion of (x-y)^7
The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. We can apply this theorem to expand (x-y)^7.
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) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
- ∑_(k=0)^n represents the summation from k = 0 to k = n.
Expanding (x - y)^7
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Set up the expansion:
We'll use the binomial theorem with n = 7. We need to find the terms of the form (7 choose k) * x^(7-k) * (-y)^k for each value of k from 0 to 7.
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Calculate the binomial coefficients:
- (7 choose 0) = 1
- (7 choose 1) = 7
- (7 choose 2) = 21
- (7 choose 3) = 35
- (7 choose 4) = 35
- (7 choose 5) = 21
- (7 choose 6) = 7
- (7 choose 7) = 1
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Write out the terms:
- k = 0: (7 choose 0) * x^(7-0) * (-y)^0 = x^7
- k = 1: (7 choose 1) * x^(7-1) * (-y)^1 = -7x^6y
- k = 2: (7 choose 2) * x^(7-2) * (-y)^2 = 21x^5y^2
- k = 3: (7 choose 3) * x^(7-3) * (-y)^3 = -35x^4y^3
- k = 4: (7 choose 4) * x^(7-4) * (-y)^4 = 35x^3y^4
- k = 5: (7 choose 5) * x^(7-5) * (-y)^5 = -21x^2y^5
- k = 6: (7 choose 6) * x^(7-6) * (-y)^6 = 7xy^6
- k = 7: (7 choose 7) * x^(7-7) * (-y)^7 = -y^7
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Combine the terms:
(x - y)^7 = x^7 - 7x^6y + 21x^5y^2 - 35x^4y^3 + 35x^3y^4 - 21x^2y^5 + 7xy^6 - y^7
Key Observations
- The coefficients alternate in sign due to the (-y) term.
- The exponents of x decrease from 7 to 0, while the exponents of y increase from 0 to 7.
- The sum of the exponents in each term always equals 7.
The expanded form of (x-y)^7 helps us understand how the terms combine and how the powers of x and y interact. This expansion is useful in various areas of mathematics, including algebra, calculus, and probability.