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Chapter 0: Problem 7

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

Short Answer

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

Step by step solution

01

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: \(\frac{3(x-3)}{(x-3)^{2}}\)
02

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 \(\frac{3}{x-3}\)
03

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.

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