Find the constant term in the expansion of (3x - 4/x)^5.

Find the constant term in the expansion of (3x - 4/x)^5.

Step 1: Identify the expression to expand, which is (3x - 4/x)^5.
Step 2: Use the Binomial Theorem to expand the expression. The general term in the expansion is given by T(k) = nCk * (a)^(n-k) * (b)^k, where n is the power, a is the first term, b is the second term, and k is the term number.
Step 3: In our case, n = 5, a = 3x, and b = -4/x.
Step 4: We need to find the term where the power of x is zero. This happens when the exponent of x in (3x)^(n-k) and (-4/x)^k cancels out.
Step 5: The term (3x)^(n-k) contributes (3^(n-k) * x^(n-k)) and (-4/x)^k contributes (-4^k * x^(-k)).
Step 6: Set the total power of x to zero: (n-k) - k = 0, which simplifies to n = 2k. Since n = 5, we have 5 = 2k, so k = 2.5. This is not an integer, so we check k = 2 and k = 3.
Step 7: For k = 2, we have n-k = 3. The term is T(2) = 5C2 * (3x)^3 * (-4/x)^2.
Step 8: Calculate 5C2 = 10, (3x)^3 = 27x^3, and (-4/x)^2 = 16/x^2.
Step 9: Combine these: T(2) = 10 * 27x^3 * 16/x^2 = 10 * 27 * 16 * x^(3-2) = 4320x.
Step 10: For k = 3, we have n-k = 2. The term is T(3) = 5C3 * (3x)^2 * (-4/x)^3.
Step 11: Calculate 5C3 = 10, (3x)^2 = 9x^2, and (-4/x)^3 = -64/x^3.
Step 12: Combine these: T(3) = 10 * 9x^2 * (-64/x^3) = -5760/x.
Step 13: The constant term occurs when the power of x is zero, which we found in T(3). Therefore, the constant term is -5760.

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 binomial coefficients (nCr) to determine the number of ways to choose terms in the expansion.