In the expansion of (x + 2)^5, what is the coefficient of x^4?

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Question: In the expansion of (x + 2)^5, what is the coefficient of x^4?

Options:

Correct Answer: 10

Solution:

The coefficient of x^4 is given by C(5, 4)(2)^1 = 5 * 2 = 10.

In the expansion of (x + 2)^5, what is the coefficient of x^4?

In the expansion of (x + 2)^5, what is the coefficient of x^4?

Step 1: Identify the expression to expand, which is (x + 2)^5.
Step 2: Recognize that we need 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 = 5.
Step 5: We want the term where x is raised to the power of 4, which means we need k = 1 (since 5 - 4 = 1).
Step 6: Calculate the binomial coefficient C(5, 1), which is the number of ways to choose 1 from 5. This equals 5.
Step 7: Calculate 2 raised to the power of 1, which is 2.
Step 8: Multiply the results from Step 6 and Step 7: 5 * 2 = 10.
Step 9: Conclude that the coefficient of x^4 in the expansion of (x + 2)^5 is 10.

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 – Understanding how to determine the power of each term in the expansion and how it relates to the coefficients.