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Find the coefficient of x^2 in the expansion of (x - 5)^5.
Find the coefficient of x^2 in the expansion of (x - 5)^5.
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Q1
Find the coefficient of x^2 in the expansion of (x - 5)^5.
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The coefficient of x^2 is C(5,2) * (-5)^3 = 10 * (-125) = -1250.
Questions & Step-by-step Solutions
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Q
Q: Find the coefficient of x^2 in the expansion of (x - 5)^5.
Solution:
The coefficient of x^2 is C(5,2) * (-5)^3 = 10 * (-125) = -1250.
Steps: 9
Show Steps
Step 1: Identify the expression we need to expand, which is (x - 5)^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 = x, b = -5, and n = 5.
Step 4: We want the coefficient of x^2, which means we need to find the term where the power of x is 2.
Step 5: To find this term, we set n-k = 2, which means k = 5 - 2 = 3.
Step 6: Calculate the binomial coefficient C(5, 3), which is the number of ways to choose 3 items from 5. C(5, 3) = 5! / (3! * (5-3)!) = 10.
Step 7: Now, we need to calculate (-5)^3, which is -5 * -5 * -5 = -125.
Step 8: Multiply the coefficient C(5, 3) by (-5)^3: 10 * (-125) = -1250.
Step 9: The coefficient of x^2 in the expansion of (x - 5)^5 is -1250.
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