Problem 8 - College Algebra Essentials

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{4 x-8}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(4 / (x - 2)\), and the excluded value from its domain is 2.

Step 1: Factor the Numerator and Denominator

First, factor out the numerator and denominator of the rational expression. In this case, we factor out a 4 from the numerator: \(4(x-2)\), and we see that the denominator is a perfect square trinomial: \((x-2)^2\).

Step 2: Simplify the Rational Expression

When simplifying rational expressions, cancel out common factors that appear in both the numerator and denominator. Here, we have a common factor of \((x-2)\). We cancel out this common factor to obtain the simplified expression: \(4 / (x-2)\).

Step 3: Find the Excluded Values

The excluded values are those that make the denominator of the simplified expression equal to zero, as dividing by zero is undefined. Solve \(x - 2 = 0\) to obtain \(x = 2\). Therefore, 2 is excluded from the domain of the simplified rational expression.