Calculate the coefficient of x^4 in the expansion of (x + 2)^6.

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Question: Calculate the coefficient of x^4 in the expansion of (x + 2)^6.

Options:

Correct Answer: 90

Solution:

The coefficient of x^4 is C(6,4) * 2^2 = 15 * 4 = 60.

Calculate the coefficient of x^4 in the expansion of (x + 2)^6.

Calculate the coefficient of x^4 in the expansion of (x + 2)^6.

Step 1: Identify the expression we need to expand, which is (x + 2)^6.
Step 2: Understand that we want to find the coefficient of x^4 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 = 2, and n = 6.
Step 5: We need to find the term where x is raised to the power of 4. This means we need k = 2 (because 6 - 4 = 2).
Step 6: 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 7: Calculate 2^2, which is the value of b raised to the power of k. 2^2 = 4.
Step 8: Multiply the results from Step 6 and Step 7 to find the coefficient of x^4: 15 * 4 = 60.

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.
Power of a Term – Identifying the specific term in the expansion that corresponds to a given power of x.