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Expanding (x + y)^4: A Detailed Exploration
The expression (x + y)^4 represents the fourth power of the binomial (x + y). Expanding this expression can be done using the Binomial Theorem or by repeated multiplication.
Understanding the Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (x + y)^n, where 'n' is a non-negative integer. The formula is:
(x + y)^n = ∑ (n choose k) * x^(n-k) * y^k
where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
- k ranges from 0 to n.
Applying the Binomial Theorem to (x + y)^4
Using the Binomial Theorem, we can expand (x + y)^4 as follows:
(x + y)^4 = (4 choose 0) * x^4 * y^0 + (4 choose 1) * x^3 * y^1 + (4 choose 2) * x^2 * y^2 + (4 choose 3) * x^1 * y^3 + (4 choose 4) * x^0 * y^4
Calculating the binomial coefficients:
- (4 choose 0) = 1
- (4 choose 1) = 4
- (4 choose 2) = 6
- (4 choose 3) = 4
- (4 choose 4) = 1
Substituting these values back into the equation:
(x + y)^4 = 1 * x^4 + 4 * x^3 * y + 6 * x^2 * y^2 + 4 * x * y^3 + 1 * y^4
Therefore, the expanded form of (x + y)^4 is:
x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
Expanding through Repeated Multiplication
Alternatively, we can expand (x + y)^4 by repeated multiplication:
(x + y)^4 = (x + y) * (x + y) * (x + y) * (x + y)
We can first expand the first two factors:
(x + y) * (x + y) = x^2 + 2xy + y^2
Then, multiply this result by (x + y) again:
(x^2 + 2xy + y^2) * (x + y) = x^3 + 3x^2y + 3xy^2 + y^3
Finally, multiply this result by (x + y) one more time:
(x^3 + 3x^2y + 3xy^2 + y^3) * (x + y) = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
This result matches the one obtained using the Binomial Theorem.
Conclusion
Expanding (x + y)^4 using the Binomial Theorem or repeated multiplication demonstrates the power of these mathematical tools for simplifying complex expressions. The resulting expanded form provides a clear understanding of the relationship between the variables x and y in this fourth-power expression.