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In the expansion of (2x - 3)^5, what is the coefficient of x^4?
In the expansion of (2x - 3)^5, what is the coefficient of x^4?
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Q1
In the expansion of (2x - 3)^5, what is the coefficient of x^4?
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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: In the expansion of (2x - 3)^5, what is the coefficient of x^4?
Solution:
The coefficient of x^4 is C(5,4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Steps: 11
Show Steps
Step 1: Identify the expression we are expanding, which is (2x - 3)^5.
Step 2: Recognize that we need to find the coefficient of x^4 in this expansion.
Step 3: 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 4: In our case, a = 2x, b = -3, and n = 5.
Step 5: We want the term where the power of x is 4. This happens when k = 1 (because 5 - k = 4).
Step 6: Calculate C(5, 1), which is the number of ways to choose 1 from 5. C(5, 1) = 5.
Step 7: Calculate (2)^4, which is 16.
Step 8: Calculate (-3)^1, which is -3.
Step 9: Multiply these values together: 5 (from C(5, 1)) * 16 (from (2)^4) * (-3) (from (-3)^1).
Step 10: Perform the multiplication: 5 * 16 = 80, and then 80 * (-3) = -240.
Step 11: Conclude that the coefficient of x^4 in the expansion of (2x - 3)^5 is -240.
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