In the expansion of (x + 2)^8, what is the coefficient of x^6?

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Question: In the expansion of (x + 2)^8, what is the coefficient of x^6?

Options:

Correct Answer: 84

Solution:

The coefficient of x^6 in (x + 2)^8 is C(8, 6) * (2)^2 = 28 * 4 = 112.

In the expansion of (x + 2)^8, what is the coefficient of x^6?

In the expansion of (x + 2)^8, what is the coefficient of x^6?

Step 1: Identify the expression we are working with, which is (x + 2)^8.
Step 2: Recognize that we want to find the coefficient of x^6 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 = 8.
Step 5: We need to find the term where x is raised to the power of 6. This means we need k = 2 (since 8 - 6 = 2).
Step 6: Calculate C(8, 2), which is the number of ways to choose 2 items from 8. C(8, 2) = 8! / (2!(8-2)!) = 28.
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: 28 * 4 = 112.

Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose terms from the expansion.
Powers of a Constant – Understanding how to calculate powers of constants (in this case, 2) in the expansion.