%5E4-expand.jpeg "(x+y)^4 Expand")
Expanding (x + y)^4
The expression (x + y)^4 represents the product of (x + y) multiplied by itself four times. Expanding this expression can be done in a few ways, but the most common and efficient method is using the Binomial Theorem.
The Binomial Theorem
The Binomial Theorem provides a general formula for expanding any power of a binomial:
(x + y)^n = x^n + nC1 * x^(n-1) * y + nC2 * x^(n-2) * y^2 + ... + nCn-1 * x * y^(n-1) + y^n
where nC1, nC2, etc. represent the binomial coefficients, calculated as follows:
nCk = n! / (k! * (n-k)!)
Expanding (x + y)^4 using the Binomial Theorem
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Identify n: In our case, n = 4.
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Apply the Binomial Theorem:
(x + y)^4 = x^4 + 4C1 * x^3 * y + 4C2 * x^2 * y^2 + 4C3 * x * y^3 + y^4
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Calculate the binomial coefficients:
- 4C1 = 4! / (1! * 3!) = 4
- 4C2 = 4! / (2! * 2!) = 6
- 4C3 = 4! / (3! * 1!) = 4
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Substitute the coefficients and simplify:
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
Other Methods for Expansion
While the Binomial Theorem provides the most concise solution, you can also expand (x + y)^4 through repeated multiplication:
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(x + y)^2 = (x + y)(x + y) = x^2 + 2xy + y^2
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(x + y)^4 = (x + y)^2 * (x + y)^2 = (x^2 + 2xy + y^2)(x^2 + 2xy + y^2)
Expanding this product will lead to the same result as using the Binomial Theorem.
Understanding the Result
The expansion of (x + y)^4 showcases the Pascal's Triangle pattern, which provides a visual representation of the binomial coefficients. Each row of Pascal's Triangle represents the coefficients for a specific power of (x + y).
The expanded form of (x + y)^4 is crucial in various mathematical and scientific applications, particularly when dealing with polynomials, probability, and series expansions.