(2×+3y)^4 using binomial expansion

Answer 1

To expand (2x + 3y)^4 using the binomial expansion, we can use the following formula:

(2x + 3y)^4 = C(4,0)(2x)^4(3y)^0 + C(4,1)(2x)^3(3y)^1 + C(4,2)(2x)^2(3y)^2 + C(4,3)(2x)^1(3y)^3 + C(4,4)(2x)^0(3y)^4

Where C(n, k) represents the binomial coefficient ("n choose k").

Calculating the coefficients:

C(4,0) = 1
C(4,1) = 4
C(4,2) = 6
C(4,3) = 4
C(4,4) = 1

Expanding and simplifying each term:

(2x + 3y)^4 = 1(2x)^4(3y)^0 + 4(2x)^3(3y)^1 + 6(2x)^2(3y)^2 + 4(2x)^1(3y)^3 + 1(2x)^0(3y)^4

= 1 * (2^4 * x^4 * 3^0) + 4 * (2^3 * x^3 * 3^1) + 6 * (2^2 * x^2 * 3^2) + 4 * (2^1 * x^1 * 3^3) + 1 * (2^0 * x^0 * 3^4)

= 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4

The expansion of (2x + 3y)^4 is 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4.