%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:
(x-y)^4 = (x-y) * (x-y) * (x-y) * (x-y)
Expanding this expression can be done in a few ways:
1. Using the Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (x + y)^n:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
where (n choose k) represents the binomial coefficient, calculated as n!/(k!*(n-k)!).
For (x-y)^4, we apply the Binomial Theorem with n = 4 and y = -y:
(x - y)^4 = ∑_(k=0)^4 (4 choose k) * x^(4-k) * (-y)^k
Expanding this summation:
(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 and simplifying:
(x-y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4
2. Repeated Multiplication
We can expand (x-y)^4 by multiplying the terms step-by-step:
- (x-y)^2 = (x-y)*(x-y) = x^2 - 2xy + y^2
- (x-y)^3 = (x-y)^2 * (x-y) = (x^2 - 2xy + y^2) * (x-y) = x^3 - 3x^2y + 3xy^2 - y^3
- (x-y)^4 = (x-y)^3 * (x-y) = (x^3 - 3x^2y + 3xy^2 - y^3) * (x-y) = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4
This method can be more time-consuming, but it illustrates the process of repeated multiplication.
3. Pascal's Triangle
Pascal's Triangle provides a visual representation of binomial coefficients. Each row corresponds to a power of (x+y), and the numbers in the row represent the coefficients in the expansion.
For (x-y)^4, we look at the fifth row of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
The coefficients are 1, 4, 6, 4, and 1. We can use these coefficients, along with the alternating signs from the (-y) terms, to get the expansion:
(x-y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4
Conclusion
Expanding (x-y)^4 can be done using the Binomial Theorem, repeated multiplication, or Pascal's Triangle. Each method provides a systematic approach to obtain the same result:
(x-y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4