Problem 9 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{x-6}{4}\). The number that must be excluded from the domain is 6.

Step 1: Factorize the Numerator and Denominator

Factorize the numerator into \( (x-6)^{2} \) and the denominator into \(4(x-6)\). So, the expression becomes \[\frac{(x-6)^{2}}{4(x-6)}\]

Step 2: Simplify the Rational Expression

Cancel out \( (x-6) \) from the numerator and the denominator. After canceling out, the rational function simplifies to: \[\frac{x-6}{4}\]

Step 3: Find the Exclusions of the Domain

Set the original denominator \(4(x-6)\) equal to zero and solve for x to find the exclusions of the domain. \(4(x-6) = 0\) implies \(x=6\). Hence, 6 is the only number that is excluded from the domain. This is because if you substitute 6 into the original denominator, it will result in a division by zero, which is undefined in mathematics.