Problem 4 - College Algebra

Question

Find all numbers that must be excluded from the domain of each rational expression. $$\frac{x+7}{x^{2}-49}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the given rational expression are -7 and 7.

Step 1: Setting the denominator equal to zero

Set the denominator equal to zero: \( x^{2}-49 = 0 \).

Step 2: Solve for x

Now, apply the difference of squares to facilitate the solution: \( (x+7)(x-7) = 0 \). Then, the solutions to the equation are given by the roots of each factor: \( x+7=0 \) and \( x-7=0 \). Solving these equations, we find \( x=-7, 7 \).

Step 3: Conclusion for the domain

The numbers -7 and 7 make the denominator equal to zero, thus they should be excluded from the domain of the given rational expression. So, the domain consists of all real numbers except -7 and 7.