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)/(x^2−6x+9)

Verified step by step guidance

1. Factor the numerator and the denominator. The numerator 3x - 9 can be factored as 3(x - 3). The denominator x^2 - 6x + 9 can be factored as (x - 3)(x - 3) or (x - 3)^2.
2. Simplify the rational expression by canceling out common factors. The factor (x - 3) in the numerator and one of the (x - 3) in the denominator can be canceled out.
3. Write the simplified expression. After canceling, the expression simplifies to 3/(x - 3).
4. Determine the values to be excluded from the domain. Since division by zero is undefined, set the denominator equal to zero and solve for x. This gives x - 3 = 0, so x = 3 must be excluded from the domain.
5. State the simplified expression and the excluded value. The simplified expression is 3/(x - 3), and the value x = 3 must be excluded from the domain to avoid division by zero.

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 that may arise from the denominator.

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 crucial for simplifying rational expressions, as it allows us to cancel common factors in the numerator and denominator.

Domain Restrictions

The domain of a rational expression consists of all possible values of the variable that do not make the denominator equal to zero. Identifying these restrictions is essential, as any value that causes the denominator to be zero must be excluded from the domain to ensure the expression is defined.