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Chapter 0: Problem 9

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}-12 x+36}{4 x-24}$$

Short Answer

Expert verified
The required simplified form of the rational expression is \(\frac{x-6}{4}\). The number that should be excluded from the domain is \(x = 6\)

Step by step solution

01

Factor the Numerator and the Denominator

Factor the numerator \((x^2-12x+36)\) and the denominator \((4x-24)\)\n\nThe factored version of the numerator is \((x-6)^2\) and of the denominator is \(4(x-6)\)
02

Simplifying the Expression

Now, simplify the expression by canceling the common factors from the numerator and the denominator, resulting in \(\frac{x-6}{4}\)
03

Identify Numbers to Exclude from the Domain

The original denominator is \(4x - 24\). Setting this equal to zero produces the equation \(4x - 24 = 0\).\n Solving this equation gives the excluded number from the domain. Solve for \(x\) to get \(x = 6\). This is the number to exclude from the domain, because it makes the denominator of the original expression equal to zero.

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