Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

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Question: Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

Options:

Correct Answer: 108

Solution:

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

Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

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 the coefficient of x^4, which means we need to find the term where the power of x is 4.
Step 5: To find this term, we set n - k = 4, which means k = 6 - 4 = 2.
Step 6: Calculate C(6, 2), which is the number of ways to choose 2 from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 7: Now calculate 3^k, which is 3^2 = 9.
Step 8: Multiply the coefficient C(6, 2) by 3^2: 15 * 9 = 135.
Step 9: The coefficient of x^4 in the expansion of (x + 3)^6 is 135.

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