In Exercises 15–58, find each product. (x−1)^3

Verified step by step guidance

1. Recognize that \((x-1)^3\) is a binomial raised to the third power, which can be expanded using the binomial theorem.
2. The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = x\), \(b = -1\), and \(n = 3\).
3. Apply the binomial theorem: \((x-1)^3 = \binom{3}{0} x^3 (-1)^0 + \binom{3}{1} x^2 (-1)^1 + \binom{3}{2} x^1 (-1)^2 + \binom{3}{3} x^0 (-1)^3\).
4. Calculate each term: \(\binom{3}{0} x^3 (-1)^0 = x^3\), \(\binom{3}{1} x^2 (-1)^1 = -3x^2\), \(\binom{3}{2} x^1 (-1)^2 = 3x\), \(\binom{3}{3} x^0 (-1)^3 = -1\).
5. Combine all the terms to get the expanded form: \(x^3 - 3x^2 + 3x - 1\).

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion is achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This concept is essential for expanding polynomials like (x - 1)^3.

Cubic Functions

Cubic functions are polynomial functions of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d. Understanding cubic functions is crucial for recognizing the behavior of the graph, including its turning points and intercepts. In the context of the question, expanding (x - 1)^3 results in a cubic polynomial.

Factoring and Simplifying Polynomials

Factoring and simplifying polynomials involves rewriting a polynomial as a product of its factors, which can make it easier to analyze or solve. This process often includes identifying common factors or applying special product formulas. In the case of (x - 1)^3, recognizing it as a repeated factor allows for straightforward expansion and simplification.