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

Step by step solution

01

Simplify the rational expression

Both the numerator and the denominator can be factored further. In the numerator, 3 can be factored out, while in the denominator, the trinomial factorization is applied. Simplifying \(\frac{3 x - 9}{x^{2} - 6x + 9}\) leads to \(\frac{3(x - 3)}{(x - 3)^{2}}\). The numerator and one factor from the denominator cancel out, leading to the simplified expression \(\frac{3}{x - 3}\).
02

Identify the excluded values

The denominator cannot be zero, so excluded values from the domain will occur when \(x - 3 = 0\). Solving for x, simple algebra leads to \(x = 3\). So, \(x = 3\) is an excluded value from the domain of the simplified expression.
03

State the Simplified Expression and the Excluded Value

After simplification, the expression \(\frac{3}{x - 3}\) remains. Thus, the number that must be excluded from the domain of this expression is \(x = 3\).

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Most popular questions from this chapter

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