Problem 14 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}-14 x+49}{x^{2}-49}$$

Step-by-Step Solution

Answer

The simplified expression is \((x-7)/(x+7)\) and the number to be excluded from the domain is \(-7\).

Step 1: Simplify the Rational Expression

Rewrite the numerator \(x^{2}-14x+49\) as \((x-7)^2\), and the denominator \(x^{2}-49\) as \((x-7)(x+7)\). As such, the rational expression can be rewritten as \((x-7)/(x+7)\) because one '(x-7)' term from the numerator can cancel out one '(x-7)' term from the denominator.

Step 2: Find the Excluded Numbers

Rational expressions are undefined when the denominator equals zero. Therefore, set the denominator equal to zero and solve for x: \(x+7=0\) which gives \(x=-7\).