Problem 7 - College Algebra Essentials

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

The simplified rational expression is \(\frac{3}{x - 3}\) and the value excluded from the domain is \(x = 3\).

Step 1: Simplify the rational expression

To simplify the rational expression, first factoring out \(3\) from the numerator and the terms in the denominator to simplify the expression. The expression becomes \(\frac{3(x - 3)}{(x-3)^{2}}\). We can cancel out the \((x - 3)\) term in the numerator and denominator. The simplified expression is then \(\frac{3}{x - 3}\).

Step 2: Find the domain of the simplified expression

Now we find out which values of \(x\) are to be excluded from the domain of the expression. By setting the denominator of the simplified rational expression equal to zero and solving for \(x\), we get \(x - 3 = 0\), which implies that \(x = 3\). Thus, \(x = 3\) must be excluded from the domain as it would make the denominator equal to zero and the expression undefined.