In the expansion of (x + 2)^6, what is the term containing x^4?

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Question: In the expansion of (x + 2)^6, what is the term containing x^4?

Options:

Correct Answer: 720

Solution:

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

In the expansion of (x + 2)^6, what is the term containing x^4?

In the expansion of (x + 2)^6, what is the term containing x^4?

Step 1: Identify the expression we are expanding, which is (x + 2)^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 = 2, and n = 6.
Step 4: We want the term that contains x^4. This means we need to find the term where the exponent of x is 4.
Step 5: If x has an exponent of 4, then 2 must have an exponent of (6 - 4) = 2.
Step 6: The term we are looking for is given by C(6, 4) * (x)^4 * (2)^2.
Step 7: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to 15.
Step 8: Calculate (2)^2, which is equal to 4.
Step 9: Now, combine these values: 15 * 4 * (x)^4.
Step 10: Multiply 15 and 4 to get 60, so the term is 60x^4.

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.
Term Extraction – Identifying specific terms in the expansion based on the powers of the variables.