%5E4.jpeg "(x-2y)^4")
Expanding (x - 2y)^4
The expression (x - 2y)^4 represents the product of (x - 2y) multiplied by itself four times:
(x - 2y)^4 = (x - 2y)(x - 2y)(x - 2y)(x - 2y)
There are several ways to expand this expression:
1. Using the Binomial Theorem
The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. In our case, a = x, b = -2y, and n = 4.
The formula is:
(a + b)^n = Σ (n choose k) a^(n-k) b^k
where (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!), and the summation runs from k = 0 to k = n.
Applying this to our problem:
(x - 2y)^4 = Σ (4 choose k) x^(4-k) (-2y)^k
Expanding the summation:
(x - 2y)^4 = (4 choose 0) x^4 (-2y)^0 + (4 choose 1) x^3 (-2y)^1 + (4 choose 2) x^2 (-2y)^2 + (4 choose 3) x^1 (-2y)^3 + (4 choose 4) x^0 (-2y)^4
Calculating the binomial coefficients and simplifying:
(x - 2y)^4 = x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4
2. Using Repeated Multiplication
We can expand the expression by multiplying the factors one by one:
- (x - 2y)(x - 2y) = x^2 - 4xy + 4y^2
- (x^2 - 4xy + 4y^2)(x - 2y) = x^3 - 6x^2y + 12xy^2 - 8y^3
- (x^3 - 6x^2y + 12xy^2 - 8y^3)(x - 2y) = x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4
Therefore, (x - 2y)^4 = x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4
Both methods lead to the same result: x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4
This is the expanded form of (x - 2y)^4.