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

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

Options:

Correct Answer: -80x^2

Solution:

The term containing x^2 is given by C(6, 2) * (x)^2 * (-2)^(6-2) = 15 * x^2 * 16 = 240x^2, so the term is -80x^2.

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

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

Step 1: Identify the expression we are expanding, which is (x - 2)^6.
Step 2: Recognize that we need to find the term that contains x^2.
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 want the term where x has the power of 2, which means we need k = 6 - 2 = 4.
Step 6: Calculate C(6, 4), which is the number of combinations of 6 items taken 4 at a time. C(6, 4) = 15.
Step 7: Now, substitute into the formula: C(6, 4) * (x)^(6-4) * (-2)^4.
Step 8: This simplifies to 15 * (x)^2 * 16, since (-2)^4 = 16.
Step 9: Multiply 15 by 16 to get 240, so we have 240x^2.
Step 10: Since the term is negative due to the original expression (x - 2), the final term is -80x^2.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Combination Formula – Using the combination formula C(n, k) to find the coefficients in the expansion.
Negative Exponents – Handling negative numbers in the expansion correctly, especially when raised to a power.