Find the domain of each rational expression. x³ - 1 / x - 1

Verified step by step guidance

1. Identify the rational expression given: \(\frac{x^3 - 1}{x - 1}\).
2. Recall that the domain of a rational expression excludes values that make the denominator zero, because division by zero is undefined.
3. Set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\).
4. Solve the equation to find the value(s) to exclude from the domain: \(x = 1\).
5. Conclude that the domain is all real numbers except \(x = 1\), which can be written in interval notation as \((-\infty, 1) \cup (1, \infty)\).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding rational expressions involves knowing how to simplify, factor, and analyze these fractions, especially focusing on restrictions caused by the denominator.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational expressions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This skill helps identify values that make the denominator zero by factoring and setting each factor equal to zero to find restrictions on the domain.