%5E6-binomial-expansion.jpeg "(x-y)^6 Binomial Expansion")
The Binomial Theorem: Expanding (x-y)^6
The binomial theorem is a powerful tool that allows us to expand expressions of the form (x + y)^n, where n is a non-negative integer. This article will focus on expanding the specific case of (x - y)^6.
Understanding the Binomial Theorem
The binomial theorem states:
(x + y)^n = Σ (n choose k) * x^(n-k) * y^k, where k goes from 0 to n.
Here:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This represents the number of ways to choose k items from a set of n items.
- Σ signifies the sum of all terms from k = 0 to k = n.
Expanding (x - y)^6
Let's apply the binomial theorem to expand (x - y)^6.
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Identify n: In this case, n = 6.
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Calculate the Binomial Coefficients: We'll need the binomial coefficients for k = 0 to 6. You can use Pascal's Triangle or the formula to calculate these:
- (6 choose 0) = 1
- (6 choose 1) = 6
- (6 choose 2) = 15
- (6 choose 3) = 20
- (6 choose 4) = 15
- (6 choose 5) = 6
- (6 choose 6) = 1
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Apply the Binomial Theorem: We substitute the values into the formula, remembering that y is negative:
- (x - y)^6 = (6 choose 0) * x^6 * (-y)^0 + (6 choose 1) * x^5 * (-y)^1 + (6 choose 2) * x^4 * (-y)^2 + (6 choose 3) * x^3 * (-y)^3 + (6 choose 4) * x^2 * (-y)^4 + (6 choose 5) * x^1 * (-y)^5 + (6 choose 6) * x^0 * (-y)^6
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Simplify:
- (x - y)^6 = x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6
Conclusion
Therefore, the expanded form of (x - y)^6 is: x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6.
Understanding the binomial theorem and its applications is crucial in various fields like algebra, calculus, and probability. Mastering this concept will equip you with a valuable tool for solving a range of mathematical problems.