%5E4.jpeg "(3x-2y)^4")
Expanding (3x-2y)^4 using the Binomial Theorem
The expression (3x-2y)^4 can be expanded using the Binomial Theorem, which states:
(a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k
where (n choose k) is the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Let's break down the expansion of (3x - 2y)^4 step-by-step:
1. Identify 'a' and 'b'
In our expression, a = 3x and b = -2y.
2. Determine the value of 'n'
Here, n = 4.
3. Apply the Binomial Theorem formula
Expanding the expression using the formula:
(3x - 2y)^4 = (4 choose 0) (3x)^4 (-2y)^0 + (4 choose 1) (3x)^3 (-2y)^1 + (4 choose 2) (3x)^2 (-2y)^2 + (4 choose 3) (3x)^1 (-2y)^3 + (4 choose 4) (3x)^0 (-2y)^4
4. Calculate the binomial coefficients
- (4 choose 0) = 4! / (0! * 4!) = 1
- (4 choose 1) = 4! / (1! * 3!) = 4
- (4 choose 2) = 4! / (2! * 2!) = 6
- (4 choose 3) = 4! / (3! * 1!) = 4
- (4 choose 4) = 4! / (4! * 0!) = 1
5. Simplify the expression
Substituting the coefficients and simplifying:
(3x - 2y)^4 = 1(81x^4) + 4(27x^3)(-2y) + 6(9x^2)(4y^2) + 4(3x)(-8y^3) + 1(16y^4)
Finally, we get:
(3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4