Calculate the term independent of x in the expansion of (2x^2 - 3x)^4.

Calculate the term independent of x in the expansion of (2x^2 - 3x)^4.

Step 1: Identify the expression to expand, which is (2x^2 - 3x)^4.
Step 2: Use the binomial theorem to expand the expression. The general term in the expansion is given by C(n, k) * (first term)^(n-k) * (second term)^k, where n is the power (4 in this case) and k is the term number.
Step 3: We need to find the term that does not contain x (the term independent of x). This occurs when the powers of x cancel out.
Step 4: In the term (2x^2)^(n-k) and (-3x)^k, the power of x is given by 2(n-k) + k. We want this to equal 0 (no x).
Step 5: Set up the equation: 2(n-k) + k = 0. Substitute n = 4: 2(4-k) + k = 0.
Step 6: Simplify the equation: 8 - 2k + k = 0, which simplifies to 8 - k = 0, so k = 8. This is incorrect; we need to find k such that 2k = 4.
Step 7: Solve for k: 2k = 4 gives k = 2.
Step 8: Now substitute k = 2 back into the general term formula: C(4, 2) * (2x^2)^(4-2) * (-3x)^2.
Step 9: Calculate C(4, 2), which is 6.
Step 10: Calculate (2x^2)^2, which is 4x^4.
Step 11: Calculate (-3x)^2, which is 9x^2.
Step 12: Combine these results: 6 * 4 * 9 = 216.
Step 13: Since we are looking for the term independent of x, we take the negative sign from (-3x)^2, resulting in -216.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding the Constant Term – Identifying the term in the expansion that does not contain the variable x.
Combinatorial Coefficients – Using binomial coefficients to determine the coefficients of the terms in the expansion.