Find the term containing x^3 in the expansion of (x - 1)^5.

Find the term containing x^3 in the expansion of (x - 1)^5.

Step 1: Identify the expression to expand, which is (x - 1)^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 = -1, and n = 5.
Step 4: We want the term that contains x^3, which means we need to find k such that n - k = 3. Here, n = 5, so k = 5 - 3 = 2.
Step 5: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 items from 5. C(5, 2) = 5! / (2!(5-2)!) = 10.
Step 6: Write the term using the binomial coefficient: C(5, 2) * x^(5-2) * (-1)^2 = 10 * x^3 * 1.
Step 7: The term containing x^3 is therefore 10 * x^3.

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.
Negative Exponents – Understanding how to handle negative terms in the expansion, particularly when they affect the sign of the coefficients.