Find the coefficient of x^4 in the expansion of (x + 3)^6.

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Question: Find the coefficient of x^4 in the expansion of (x + 3)^6.

Options:

Correct Answer: 243

Solution:

The coefficient of x^4 is C(6,4) * (3)^2 = 15 * 9 = 135.

Find the coefficient of x^4 in the expansion of (x + 3)^6.

Find the coefficient of x^4 in the expansion of (x + 3)^6.

Step 1: Identify the expression to expand, which is (x + 3)^6.
Step 2: 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 3: In our case, a = x, b = 3, and n = 6.
Step 4: We want to find the term where x is raised to the power of 4, which means we need k = 2 (since 6 - 4 = 2).
Step 5: Calculate C(6, 2), which is the number of ways to choose 2 items from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 6: Calculate 3^2, which is the value of b raised to the power of k. 3^2 = 9.
Step 7: Multiply the results from Step 5 and Step 6 to find the coefficient of x^4: 15 * 9 = 135.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.
Finding Specific Coefficients – Identifying the coefficient of a specific term in a polynomial expansion.