Problem 10 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}-8 x+16}{3 x-12}$$

Step-by-Step Solution

Answer

The simplified rational function is \(\frac{x - 4}{3}\) and the excluded value from the domain is x = 4.

Step 1: Simplify the Expression

First, factor the numerator and the denominator. The numerator can be factored to \((x - 4)(x - 4)\) or \((x - 4)^2\) and the denominator to \(3(x - 4)\). Thus, the rational function becomes \(\frac{(x-4)^2}{3(x-4)}\)

Step 2: Cancel Common Terms

The common terms between the numerator and denominator can be cancelled. Here, \((x - 4)\) is common and can be cancelled out, which simplifies the function to \(\frac{x - 4}{3}\)

Step 3: Determine the Excluded Values

The denominator cannot be zero, so we'll set it to zero and solve for x: \(3(x - 4) =0\).\nUpon solving, we'll find \(x = 4\). This is an excluded value in the domain of the rational function since it'd make the denominator zero.