In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

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Question: In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

Options:

Correct Answer: 36

Exam Year: 2023

Solution:

The coefficient of x^3 is C(4,3) * (3)^1 = 4 * 3 = 12.

In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

Step 1: Identify the expression we are expanding, which is (x + 3)^4.
Step 2: Recognize that we want to find 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 = 3, and n = 4.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 1 (since n - k = 3).
Step 6: Calculate C(4, 3), which is the number of ways to choose 3 items from 4. This equals 4.
Step 7: Calculate (3)^1, which is simply 3.
Step 8: Multiply the results from Step 6 and Step 7: 4 * 3 = 12.
Step 9: Conclude that the coefficient of x^3 in the expansion of (x + 3)^4 is 12.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the Binomial Theorem, which involves combinations and powers.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using the formula C(n, k) * a^(n-k) * b^k.