Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^5.

Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^5.

Step 1: Identify the expression to expand, which is (2x^2 - 3x + 4)^5.
Step 2: Recognize that we need to find the term that does not contain 'x' (the term independent of x).
Step 3: Use the multinomial expansion formula, which allows us to expand expressions with more than two terms.
Step 4: The general term in the expansion can be written as: T = (5! / (a! b! c!)) * (2x^2)^a * (-3x)^b * (4)^c, where a + b + c = 5.
Step 5: The powers of x in the term are given by: 2a + b. We want this to equal 0 (to be independent of x).
Step 6: Set up the equation: 2a + b = 0. Since a and b must be non-negative integers, find suitable values for a and b.
Step 7: Solve for b in terms of a: b = -2a. Since b must be non-negative, a must be 0. Thus, a = 0 and b = 0.
Step 8: Calculate c using the equation a + b + c = 5: 0 + 0 + c = 5, so c = 5.
Step 9: Substitute a, b, and c back into the general term formula: T = (5! / (0! 0! 5!)) * (2x^2)^0 * (-3x)^0 * (4)^5.
Step 10: Simplify the term: T = 1 * 1 * 1 * 1024 = 1024.
Step 11: The term independent of x in the expansion is 1024.

Binomial Expansion – The process of expanding expressions that are raised to a power, typically using the binomial theorem.
Finding the Constant Term – Identifying the term in an expansion that does not contain the variable, which occurs when the powers of the variable sum to zero.
Combination of Terms – Understanding how to combine different terms from the expansion to find the specific term that is independent of the variable.