In the expansion of (2x + 5)^3, what is the coefficient of x?

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Question: In the expansion of (2x + 5)^3, what is the coefficient of x?

Options:

Correct Answer: 45

Solution:

The coefficient of x is given by 3C1 * (2)^1 * (5)^2 = 3 * 2 * 25 = 150.

In the expansion of (2x + 5)^3, what is the coefficient of x?

In the expansion of (2x + 5)^3, what is the coefficient of x?

Step 1: Identify the expression to expand, which is (2x + 5)^3.
Step 2: Recognize that we need to find the coefficient of x in the expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n can be expanded using combinations.
Step 4: In our case, a = 2x, b = 5, and n = 3.
Step 5: The general term in the expansion is given by nCk * (a)^(n-k) * (b)^k, where k is the term number.
Step 6: We want the term that contains x, which occurs when k = 1 (since we need one x).
Step 7: Calculate nCk for k = 1: 3C1 = 3.
Step 8: Calculate (2x)^(1) = 2^1 * x^1 = 2 * x.
Step 9: Calculate (5)^(2) = 5^2 = 25.
Step 10: Multiply these values together: 3C1 * (2)^1 * (5)^2 = 3 * 2 * 25.
Step 11: Perform the multiplication: 3 * 2 = 6, then 6 * 25 = 150.
Step 12: The coefficient of x in the expansion is 150.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations and powers.