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

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

Options:

Correct Answer: -243

Solution:

The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.

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

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

Step 1: Identify the expression to expand, which is (2x - 3)^5.
Step 2: Understand that we need to find the term that does not contain 'x'. This is called the term independent of x.
Step 3: Use the binomial theorem, which states that (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 4: In our case, a = 2x, b = -3, and n = 5.
Step 5: We need to find the value of k such that the power of x is zero. This happens when (2x)^(n-k) = (2x)^0, which means n-k = 0, so k = n = 5.
Step 6: Calculate the coefficient for k = 5 using the binomial coefficient C(5, 5). This is equal to 1.
Step 7: Calculate (2x)^0, which is equal to 1.
Step 8: Calculate (-3)^5, which is equal to -243.
Step 9: Multiply the results from steps 6, 7, and 8: 1 * 1 * (-243) = -243.
Step 10: Conclude that the term independent of x in the expansion is -243.

Binomial Expansion – The process of expanding 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.