In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

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Question: In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

Options:

Correct Answer: 36

Exam Year: 2020

Solution:

The coefficient of x is given by 4C1 * (2)^1 * (3)^3 = 4 * 2 * 27 = 216.

In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

Step 1: Identify the expression to expand, which is (2x + 3)^4.
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 = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = 2x, b = 3, and n = 4.
Step 5: We want the term that contains x, which occurs when k = 1 (since we need one x).
Step 6: Calculate the binomial coefficient for k = 1: 4C1 = 4.
Step 7: Calculate (2x)^(1) = 2^1 * x^1 = 2 * x.
Step 8: Calculate (3)^(3) = 3^3 = 27.
Step 9: Multiply the results from steps 6, 7, and 8: 4 * 2 * 27.
Step 10: Perform the multiplication: 4 * 2 = 8, then 8 * 27 = 216.
Step 11: Conclude that the coefficient of x in the expansion is 216.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficients – The coefficients in the expansion, represented as nCr, which determine the number of ways to choose r elements from a set of n elements.
Power of a Term – Understanding how to calculate the power of each term in the binomial expansion to find specific coefficients.