What is the coefficient of x^2 in the expansion of (x - 4)^6?

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Question: What is the coefficient of x^2 in the expansion of (x - 4)^6?

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Correct Answer: 360

Solution:

Using the binomial theorem, the coefficient of x^2 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-4, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-4)^2 = 15 * 16 = 240.

What is the coefficient of x^2 in the expansion of (x - 4)^6?

Practice Questions

What is the coefficient of x^2 in the expansion of (x - 4)^6?

Questions & Step-by-Step Solutions

What is the coefficient of x^2 in the expansion of (x - 4)^6?

Step 1: Identify the expression we need to expand, which is (x - 4)^6.
Step 2: Recognize that we will use the binomial theorem to find the coefficient of x^2.
Step 3: In the binomial theorem, we have (a + b)^n, where n is the exponent. Here, n = 6, a = x, and b = -4.
Step 4: We need to find the term that contains x^2. This corresponds to k = 2 in the binomial expansion.
Step 5: The formula for the coefficient of x^k in (a + b)^n is given by nCk * a^(n-k) * b^k.
Step 6: Calculate nCk, which is 6C2. This is calculated as 6! / (2! * (6-2)!) = 15.
Step 7: Calculate a^(n-k), which is x^(6-2) = x^4. The coefficient here is 1.
Step 8: Calculate b^k, which is (-4)^2 = 16.
Step 9: Multiply the results: Coefficient = 6C2 * (1) * (16) = 15 * 16 = 240.
Step 10: Conclude that the coefficient of x^2 in the expansion of (x - 4)^6 is 240.

Binomial Theorem – The binomial theorem provides a formula for the expansion of powers of binomials, allowing the calculation of specific coefficients in the expansion.
Combinatorics – Understanding combinations (nCk) is essential for determining the number of ways to choose k elements from n elements.
Exponentiation – Calculating powers of numbers, particularly when dealing with negative bases and ensuring correct application of exponents.

What is the coefficient of x^2 in the expansion of (x - 4)^6?

What’s inside this PDF?

What is the coefficient of x^2 in the expansion of (x - 4)^6?

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Questions & Step-by-Step Solutions

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