What is the value of the coefficient of x^0 in the expansion of (x + 5)^3?

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Question: What is the value of the coefficient of x^0 in the expansion of (x + 5)^3?

Options:

Correct Answer: 125

Solution:

The coefficient of x^0 in (x + 5)^3 is given by 3C0 * (5)^3 = 1 * 125 = 125.

What is the value of the coefficient of x^0 in the expansion of (x + 5)^3?

What is the value of the coefficient of x^0 in the expansion of (x + 5)^3?

Step 1: Understand that x^0 means we are looking for the constant term in the expansion.
Step 2: Recognize that (x + 5)^3 is a binomial expression that can be expanded using the binomial theorem.
Step 3: The binomial theorem states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 4: In our case, a = x, b = 5, and n = 3.
Step 5: To find the coefficient of x^0, we need the term where x is raised to the power of 0. This happens when k = 3 (since 3 - k = 0).
Step 6: Calculate the coefficient using the formula: nCk * b^k, where n = 3, k = 3, and b = 5.
Step 7: Calculate 3C3, which is 1 (there's only one way to choose all 3 items).
Step 8: Calculate 5^3, which is 125.
Step 9: Multiply the results from Step 7 and Step 8: 1 * 125 = 125.
Step 10: Conclude that the coefficient of x^0 in the expansion of (x + 5)^3 is 125.

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