Problem 37 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+1}{x^{2}-2 x-3}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression \(\frac{x + 1}{x^{2} - 2x - 3}\) is \(\frac{1}{x - 3}\)

Step 1: Factorize the denominator

Our first task is to factorize the quadratic trinomial in the denominator, \(x^{2} - 2x - 3\). The quadratic trinomial can be written as \((x - a)(x - b)\) where \(a\) and \(b\) are the roots of the equation. For a quadratic expression in the form \(ax^{2} + bx + c\), the roots can be found using the formula \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). In this case, the roots can be found by factoring the equation to \((x - 3)(x + 1)\).

Step 2: Cancel out common factors

Once the trinomial is fully factored, look for common factors that appear in both the numerator and the denominator. In this case, \(x + 1\) is a common factor that we can cancel out. The final simplified expression is thus \(\frac{1}{x - 3}\)

Step 3: Check your work

To verify the solution, cross-multiply to get back to the original expression. If the original expression is regained, it confirms that the simplification is correct.