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

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

Step 1: Identify the expression to expand, which is (x^2 - 3x + 2)^4.
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 like (a + b + c)^n.
Step 4: In our case, a = x^2, b = -3x, and c = 2, and n = 4.
Step 5: We need to find combinations of these terms such that the total power of x is zero.
Step 6: The term independent of x occurs when we choose 2 from (x^2), 2 from (2), and 0 from (-3x).
Step 7: Calculate the number of ways to choose these terms using the multinomial coefficient: C(4, 2, 0, 2) = C(4, 2) = 6.
Step 8: Calculate the contribution from each term: (x^2)^2 gives x^4, (-3x)^0 gives 1, and (2)^2 gives 4.
Step 9: Combine these contributions: 6 (from the coefficient) * 4 (from (2)^2) * 9 (from (-3)^2) = 216.
Step 10: Conclude that the term independent of x in the expansion is 216.

Binomial Expansion – Understanding how to expand polynomials 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 number of ways to choose terms from the expansion.