In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. (x+5)/(x²−25)

Verified step by step guidance

1. Identify the domain of a rational expression. The domain includes all real numbers except those that make the denominator equal to zero, as division by zero is undefined.
2. Write the denominator of the given rational expression: $x^2\; -\; 25$.
3. Set the denominator equal to zero to find the values that must be excluded: $x^2\; -\; 25\; =\; 0$.
4. Factor the denominator using the difference of squares formula: $x^2\; -\; 25\; =\; (x\; -\; 5)(x\; +\; 5)$.
5. Solve for the values of $x$ that make the denominator zero: $x\; -\; 5\; =\; 0$ gives $x\; =\; 5$, and $x\; +\; 5\; =\; 0$ gives $x\; =\; -5$. These are the values that must be excluded from the domain.

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, specifically where the denominator equals zero, as division by zero is 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 make the denominator zero, as these values 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 identifying values that make the denominator zero, allowing us to determine which numbers must be excluded from the domain.