(2×+3y)^4 using binomial expansion
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.
To expand the expression (2x + 3y)^4 using the binomial expansion, follow these steps:
1. Determine the number of terms in the expansion. In this case, the power of the expression is 4, so there will be 4 + 1 = 5 terms.
2. Write down the first term of the expansion. Raise the first term in the binomial expression (2x) to the power of 4: (2x)^4 = 16x^4.
3. Write down the second term of the expansion. Raise the second term in the binomial expression (3y) to the power of 0: (3y)^0 = 1.
4. Write down the remaining terms of the expansion. Move from the first term towards the last by reducing the power of (2x) by one and increasing the power of (3y) by one each time:
Term 3: (2x)^3 * (3y)^1 = 8x^3 * 3y = 24x^3y
Term 4: (2x)^2 * (3y)^2 = 4x^2 * 9y^2 = 36x^2y^2
Term 5: (2x)^1 * (3y)^3 = 2x * 27y^3 = 54xy^3
5. Combine all the terms together. The expanded form becomes: 16x^4 + 24x^3y + 36x^2y^2 + 54xy^3 + 1.
So, (2x + 3y)^4 expands to 16x^4 + 24x^3y + 36x^2y^2 + 54xy^3 + 1.