Find the coefficient of x^3 in the expansion of (x - 3)^5.

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Question: Find the coefficient of x^3 in the expansion of (x - 3)^5.

Options:

Correct Answer: -135

Solution:

The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.

Find the coefficient of x^3 in the expansion of (x - 3)^5.

Find the coefficient of x^3 in the expansion of (x - 3)^5.

Correct Answer: -90

Step 1: Identify the expression we need to expand, which is (x - 3)^5.
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 = -3, and n = 5.
Step 4: We want the coefficient of x^3, which means we need to find the term where the power of x is 3.
Step 5: To find this term, we set n - k = 3, which means k = 5 - 3 = 2.
Step 6: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 from 5.
Step 7: C(5, 2) = 5! / (2! * (5-2)!) = 10.
Step 8: Now, calculate (-3)^2, which is 9.
Step 9: Multiply the coefficient C(5, 2) by (-3)^2: 10 * 9 = 90.
Step 10: Since we are looking for the coefficient of x^3 in (x - 3)^5, the final answer is -90 because of the negative sign in (-3).

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficient Extraction – Identifying the specific coefficient of a term in a polynomial expansion.