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

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

Step 1: Identify the expression to expand, which is (2x^2 - 3x + 4)^3.
Step 2: Understand that we need to find the term that does not contain 'x' (the term independent of x).
Step 3: Use the binomial theorem to expand the expression. We will look for combinations of terms that will cancel out the 'x' powers.
Step 4: Write down the general term in the expansion: C(3, k) * (2x^2)^(3-k) * (-3x)^k * 4^(3-k), where k is the number of times we choose the -3x term.
Step 5: Calculate the terms for k = 0, 1, 2, and 3 to find the independent term.
Step 6: For k = 0: C(3,0) * (2x^2)^3 * 4^0 = 1 * 8x^6 * 1 = 8x^6 (not independent).
Step 7: For k = 1: C(3,1) * (2x^2)^2 * (-3x)^1 * 4^1 = 3 * 4x^4 * (-3x) * 4 = -36x^5 (not independent).
Step 8: For k = 2: C(3,2) * (2x^2)^1 * (-3x)^2 * 4^1 = 3 * 2x^2 * 9x^2 * 4 = 216x^4 (not independent).
Step 9: For k = 3: C(3,3) * (2x^2)^0 * (-3x)^3 * 4^0 = 1 * (-27x^3) * 1 = -27x^3 (not independent).
Step 10: Now, we need to find the constant term. We can also calculate the constant term directly: C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3.
Step 11: Calculate C(3,0)(4)^3 = 1 * 64 = 64.
Step 12: Calculate C(3,1)(-3)(4)^2 = 3 * (-3) * 16 = -144.
Step 13: Calculate C(3,2)(-3)^2(4) = 3 * 9 * 4 = 108.
Step 14: Calculate C(3,3)(-3)^3 = 1 * (-27) = -27.
Step 15: Add the results: 64 - 144 + 108 - 27 = 1.
Step 16: The term independent of x is 28.

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 C(n, k) to determine the coefficients of the terms in the expansion.