Problem 39 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{4 x-8}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression \( \frac{4 x-8}{x^{2}-4 x+4} \) is \( \frac{4}{x-2} \).

Step 1: Factor the numerator and denominator

Factorise the numerator as \(4(x-2)\). For the denominator , it's a perfect square trinomial, which can be factored as \((x-2)^{2}\)

Step 2: Cancel out common factors

Now in simplified expression \( \frac{4(x-2)}{(x-2)(x-2)} \), we have a common factor (x-2) in the numerator and the denominator. Cancel out this common factor to get \( \frac{4}{x-2} \)

Step 3: Check for further simplifications

In the simplest form \( \frac{4}{x-2} \), no further simplification is possible as now there are no common factors left in the numerator and the denominator. Hence, the provided rational expression is fully simplified.