Find the domain of each rational expression. See Example 1. x³ - 1———— x - 1

Verified step by step guidance

1. Identify the rational expression: \( \frac{x^3 - 1}{x - 1} \).
2. Recognize that the domain of a rational expression is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero to find the values to exclude: \( x - 1 = 0 \).
4. Solve the equation \( x - 1 = 0 \) to find the value of \( x \) that makes the denominator zero.
5. Exclude the value found in the previous step from the domain of the rational expression.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for determining their domain, as the values that make the denominator zero must be excluded from the domain. In this case, the expression x³ - 1 / (x - 1) is a rational expression that requires careful analysis of its denominator.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational expressions, the domain is restricted by the values that make the denominator zero, as division by zero is undefined. Therefore, identifying these values is essential for accurately determining the domain of the given rational expression.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. In the context of the given expression, factoring x³ - 1 can help identify any common factors with the denominator, which may simplify the expression and clarify the domain. Recognizing these factors is key to understanding the behavior of the rational expression.