Find the coefficient of x^2 in the expansion of (3x - 2)^5.

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Question: Find the coefficient of x^2 in the expansion of (3x - 2)^5.

Options:

Correct Answer: -90

Solution:

The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.

Find the coefficient of x^2 in the expansion of (3x - 2)^5.

Find the coefficient of x^2 in the expansion of (3x - 2)^5.

Step 1: Identify the expression we need to expand, which is (3x - 2)^5.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 3x, b = -2, and n = 5.
Step 4: We want the coefficient of x^2, so we need to find the term where (3x) is raised to the power of 2.
Step 5: This means we need k = 3 because (n-k) = 2 (since 5 - k = 2).
Step 6: Calculate the binomial coefficient 5C3, which is equal to 5! / (3! * (5-3)!) = 10.
Step 7: Calculate (3x)^2, which is (3^2) * (x^2) = 9x^2.
Step 8: Calculate (-2)^3, which is -2 * -2 * -2 = -8.
Step 9: Multiply the results: 5C3 * (3x)^2 * (-2)^3 = 10 * 9 * (-8).
Step 10: Calculate 10 * 9 = 90, then 90 * (-8) = -720.
Step 11: The coefficient of x^2 in the expansion is -720.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Coefficient Extraction – Identifying and calculating specific coefficients from the expanded form of a binomial expression.
Combinatorics – Using combinations (nCr) to determine the number of ways to choose terms in the expansion.