(2x-3y)^4 Expand

Jasonbradley

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Jun 16, 2024

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Expanding (2x - 3y)^4 using the Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. In this case, we want to expand (2x - 3y)^4.

The Binomial Theorem:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

where:

• n is a positive integer (the power)
• k is an integer from 0 to n
• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying the Theorem to (2x - 3y)^4:

1. Identify 'a' and 'b':

   • a = 2x
   • b = -3y

2. Apply the Binomial Theorem: (2x - 3y)^4 = Σ (4 choose k) * (2x)^(4-k) * (-3y)^k

3. Expand the summation: (2x - 3y)^4 = (4 choose 0) * (2x)^4 * (-3y)^0 + (4 choose 1) * (2x)^3 * (-3y)^1 + (4 choose 2) * (2x)^2 * (-3y)^2 + (4 choose 3) * (2x)^1 * (-3y)^3 + (4 choose 4) * (2x)^0 * (-3y)^4

4. Calculate the binomial coefficients and simplify: (2x - 3y)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * (-3y) + 6 * 4x^2 * 9y^2 + 4 * 2x * (-27y^3) + 1 * 1 * 81y^4

5. Combine the terms: (2x - 3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4

Therefore, the expansion of (2x - 3y)^4 is 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4.