Problem 5 - College Algebra Essentials

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 -1 and -10 need to be excluded from the domain of the given rational expression.

Step 1: Identify the Denominator

The denominator of the given rational function is \(x^{2}+11x+10\). To find the excluded values, this expression must be set equal to zero, as you search for values for which the denominator would become zero.

Step 2: Solve for X

To find the roots of this quadratic equation you can factorise it: \(x^{2}+11x+10 = 0 \) becomes \((x + 1)(x + 10) = 0\). Then set each factor equal to zero and solve for x: \(x + 1 = 0\) gives \(x = -1\) and \(x + 10 = 0\) gives \(x = -10\)

Step 3: State the Excluded Values

The solution for this equation represents the values which would make the denominator become zero. As division by zero is undefined, these values must be excluded from the domain of the rational expression. Thus, the numbers -1 and -10 need to be excluded from the domain of the given rational expression.