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

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

Options:

Correct Answer: -12x^2

Solution:

The term containing x^2 is given by C(4, 2)(-2)^2x^2 = 6 * 4 * x^2 = 24x^2, but since it is negative, it is -12x^2.

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

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

Step 1: Identify the expression we are expanding, which is (x - 2)^4.
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 = 4.
Step 4: We want the term that contains x^2. This occurs when n-k = 2, which means k = 4 - 2 = 2.
Step 5: Calculate the binomial coefficient C(4, 2), which is the number of ways to choose 2 items from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 6: Now calculate (-2)^2, which is 4.
Step 7: Combine these results to find the term: C(4, 2) * (-2)^2 * x^2 = 6 * 4 * x^2 = 24x^2.
Step 8: Since the original expression has a negative sign in front of 2, the term is actually -24x^2.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Combination Coefficient – Using the combination formula C(n, k) to find the coefficients of the terms in the expansion.
Sign of Terms – Recognizing how the sign of terms changes based on the binomial components.