Find the constant term in the expansion of (x - 2/x)^6. (2022)

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Question: Find the constant term in the expansion of (x - 2/x)^6. (2022)

Options:

Correct Answer: -64

Solution:

The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.

Find the constant term in the expansion of (x - 2/x)^6. (2022)

Find the constant term in the expansion of (x - 2/x)^6. (2022)

Step 1: Identify the expression to expand, which is (x - 2/x)^6.
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 exponent and k is the term number.
Step 3: In our case, n = 6, the first term is x, and the second term is -2/x.
Step 4: The general term (T_k) can be written as T_k = C(6, k) * (x)^(6-k) * (-2/x)^k.
Step 5: Simplify the term: T_k = C(6, k) * (x)^(6-k) * (-2)^k * (1/x)^k = C(6, k) * (-2)^k * (x)^(6-2k).
Step 6: To find the constant term, we need the power of x to be zero. Set the exponent 6 - 2k = 0.
Step 7: Solve for k: 6 - 2k = 0 leads to 2k = 6, so k = 3.
Step 8: Substitute k = 3 into the general term to find the constant term: T_3 = C(6, 3) * (-2)^3.
Step 9: Calculate C(6, 3), which is 20, and (-2)^3, which is -8.
Step 10: Multiply these values: 20 * -8 = -160.
Step 11: The constant term in the expansion is -160.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding Constant Terms – Identifying the term in the expansion where the variable's exponent is zero.
Combinatorics – Using combinations to determine the coefficients in the binomial expansion.