Problem 7 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{3 x-9}{x^{2}-6 x+9}$$

Step-by-Step Solution

Answer

Our simplified rational expression is \(\frac{3}{x-3}\), and the value 3 must be excluded from the domain.

Step 1: Prime Factorization

Begin by simplifying the given fraction by factorizing the numerator and the denominator. The factorization will reveal common factors which can then be cancelled out. The numerator, \(3x-9\), can be factored as \(3(x-3)\). The denominator, \(x^{2}-6x+9\), is a perfect square trinomial which can be factored as \((x-3)^{2}\). Our fraction thus becomes: \(\frac{3(x-3)}{(x-3)^{2}}\)

Step 2: Cancel Out the Common Factors

Now cancel out the common factors both in the numerator and denominator. A \(x-3\) present in both the numerator and denominator can be cancelled out. Therefore, the equation simplifies to \(\frac{3}{x-3}\)

Step 3: Identify Excluded Values

Find the values for x that make the denominator zero, because these will create undefined values in the rational function. Use the equation: \(x-3 = 0\). Solving this equation for x gives us \(x = 3\). Thus, 3 is the value to be excluded from the domain of our simplified rational expression.