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In the expansion of (x + 2)^5, what is the coefficient of x^3?
In the expansion of (x + 2)^5, what is the coefficient of x^3?
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Q1
In the expansion of (x + 2)^5, what is the coefficient of x^3?
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The coefficient of x^3 is C(5,3) * (2)^2 = 10 * 4 = 40.
Questions & Step-by-step Solutions
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Q
Q: In the expansion of (x + 2)^5, what is the coefficient of x^3?
Solution:
The coefficient of x^3 is C(5,3) * (2)^2 = 10 * 4 = 40.
Steps: 9
Show Steps
Step 1: Identify the expression we are expanding, which is (x + 2)^5.
Step 2: Recognize that we want the coefficient of x^3 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 = x, b = 2, and n = 5.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 2 (since n - k = 3).
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 from 5. This is equal to 5! / (2! * (5-2)!) = 10.
Step 7: Calculate (2)^2, which is 4.
Step 8: Multiply the results from Step 6 and Step 7: 10 * 4 = 40.
Step 9: Conclude that the coefficient of x^3 in the expansion of (x + 2)^5 is 40.
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