Find the domain of each rational expression. x^3 - 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 that are not in the domain: $x-1=0$.
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 because they can have restrictions on their domain, particularly where the denominator equals zero, which would make the expression undefined.

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 excludes any values that cause the denominator to be zero, as these would lead to undefined expressions.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. This is essential for simplifying rational expressions and identifying values that make the denominator zero, thus determining the domain.