Problem 5 - College Algebra

Question

Find all numbers that must be excluded from the domain of each rational expression. $$\frac{x-1}{x^{2}+11 x+10}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the rational expression are \(x=-1\) and \(x=-10\), because these values would make the denominator equal to zero, causing the expression to be undefined.

Step 1: Set denominator equal to zero

To determine the values to exclude from the domain, set the denominator of the rational expression equal to zero. This gives you an equation of the form \(x^{2}+11 x+10=0\).

Step 2: Solve the equation

Now, solve the equation \(x^{2}+11 x+10=0\) for x. This is a quadratic equation. You can use the quadratic formula or factoring to solve this equation. In this case, the equation factors as \((x+1)(x+10)=0\).

Step 3: Find the roots

Setting each factor equal to zero gives you \(x+1=0\) and \(x+10=0\), which yields \(x=-1\) and \(x=-10\).