Binomial Formula of (x + y)^6
The binomial formula, also known as the binomial theorem, is a powerful tool used to expand expressions of the form (x + y)^n, where 'n' is a positive integer. This article will focus on expanding (x + y)^6 using the binomial formula.
Understanding the Binomial Formula
The binomial formula states:
(x + y)^n = ∑(n choose k) * x^(n-k) * y^k
where:
- n is the power to which the binomial is raised.
- k ranges from 0 to n.
- (n choose k) represents the binomial coefficient, which is the number of ways to choose k objects from a set of n objects. It is calculated as n!/(k!(n-k)!).
Expanding (x + y)^6
Let's apply the binomial formula to expand (x + y)^6:
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Identify n: n = 6
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Calculate the binomial coefficients: We need to calculate (6 choose k) for k = 0, 1, 2, 3, 4, 5, and 6.
- (6 choose 0) = 6!/(0!6!) = 1
- (6 choose 1) = 6!/(1!5!) = 6
- (6 choose 2) = 6!/(2!4!) = 15
- (6 choose 3) = 6!/(3!3!) = 20
- (6 choose 4) = 6!/(4!2!) = 15
- (6 choose 5) = 6!/(5!1!) = 6
- (6 choose 6) = 6!/(6!0!) = 1
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Apply the formula:
(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
Simplifying the expression:
(x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6
Conclusion
The binomial formula provides a systematic way to expand expressions of the form (x + y)^n. By understanding and applying the formula, we can efficiently expand these expressions, even for higher values of n. In the case of (x + y)^6, the expanded form is x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6.
