Problem 58 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+5}{x^{2}+25}$$

Step-by-Step Solution

Answer

The expression cannot be simplified further and remains as \(\frac{x+5}{x^{2}+25}\).

Step 1: Analyze the Quadratic Equation

Firstly, look for factorizing the quadratic equation which is \(x^{2}+25\). However, this equation does not have any real roots, and hence, cannot be factored in the set of real numbers.

Step 2: Looking for the Common Factors

After analysing and realising the equation in the denominator cannot be factored, check whether the terms in the numerator i.e., \(x+5\) can be factored out from both numerator and denominator. But in given situation, there are no common factors.

Step 3: Concluding statement

After the above analysis, it could be concluded that the given terms in the numerator and denominator cannot be factored or simplified any further because they don't share any common factors.