In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (3x−9)/(x²−6x+9)

Verified step by step guidance

1. Factor both the numerator and the denominator of the rational expression. The numerator, 3x - 9, can be factored as 3(x - 3). The denominator, x^2 - 6x + 9, is a perfect square trinomial and can be factored as (x - 3)(x - 3) or (x - 3)^2.
2. Rewrite the rational expression using the factored forms: (3(x - 3)) / ((x - 3)(x - 3)).
3. Identify any common factors in the numerator and denominator. In this case, (x - 3) is a common factor.
4. Cancel the common factor (x - 3) from the numerator and denominator, but note that x = 3 must be excluded from the domain because it would make the denominator zero in the original expression.
5. Write the simplified expression and state the domain restrictions. The simplified expression is 3 / (x - 3), and the domain excludes x = 3.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions involves factoring and reducing them to their simplest form, which can help identify any restrictions on the variable. Understanding how to manipulate these expressions is crucial for solving problems involving them.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original polynomial. This is essential in simplifying rational expressions, as it allows for cancellation of common factors in the numerator and denominator, leading to a more manageable form of the expression.

Domain of a Rational Expression

The domain of a rational expression consists of all the values that the variable can take without causing the denominator to equal zero. Identifying excluded values is important because these values lead to undefined expressions. In the context of the given problem, finding the domain helps ensure that the simplified expression is valid for all permissible inputs.