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Find the coefficient of x^3 in the expansion of (2x - 3)^6.
Find the coefficient of x^3 in the expansion of (2x - 3)^6.
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Practice Questions
1 question
Q1
Find the coefficient of x^3 in the expansion of (2x - 3)^6.
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The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the coefficient of x^3 in the expansion of (2x - 3)^6.
Solution:
The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Steps: 12
Show Steps
Step 1: Identify the expression to expand, which is (2x - 3)^6.
Step 2: 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 3: In our case, a = 2x, b = -3, and n = 6.
Step 4: We want the term where the power of x is 3, which means we need to find the term where (2x) is raised to the power of 3.
Step 5: This corresponds to k = 3 in the Binomial Theorem, since we want (2x)^(6-3) and (-3)^3.
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 20.
Step 7: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 8: Calculate (-3)^3, which is -3 * -3 * -3 = -27.
Step 9: Multiply these values together: 20 * 8 * (-27).
Step 10: First, calculate 20 * 8 = 160.
Step 11: Then, calculate 160 * (-27) = -4320.
Step 12: The coefficient of x^3 in the expansion is -4320.
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