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What is the coefficient of x^4 in the expansion of (2x - 3)^5?
What is the coefficient of x^4 in the expansion of (2x - 3)^5?
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Q1
What is the coefficient of x^4 in the expansion of (2x - 3)^5?
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The coefficient of x^4 is C(5, 4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Questions & Step-by-step Solutions
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Q
Q: What is the coefficient of x^4 in the expansion of (2x - 3)^5?
Solution:
The coefficient of x^4 is C(5, 4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Steps: 10
Show Steps
Step 1: Identify the expression to expand, which is (2x - 3)^5.
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 = 5.
Step 4: We want the term where the power of x is 4. This occurs when k = 1 (because 5 - k = 4).
Step 5: Calculate C(5, 1), which is the number of ways to choose 1 from 5. C(5, 1) = 5.
Step 6: Calculate (2x)^(5-1) = (2x)^4 = 2^4 * x^4 = 16 * x^4.
Step 7: Calculate (-3)^1 = -3.
Step 8: Combine these results to find the coefficient: C(5, 1) * (2^4) * (-3) = 5 * 16 * (-3).
Step 9: Perform the multiplication: 5 * 16 = 80, then 80 * (-3) = -240.
Step 10: Conclude that the coefficient of x^4 in the expansion is -240.
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