Problem 14 - College Algebra Essentials

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 \(\frac{x-7}{x+7}\) and number \(-7\) is excluded from the domain because it makes the denominator zero.

Step 1: Factorise the Numerator and Denominator

First, factorise the numerator and the denominator. The numerator \(x^{2}-14x+49\) is a perfect square trinomial that can be written as \((x-7)^{2}\) . Similarly, the denominator \(x^{2}-49\) is a difference of squares and can be factored as \((x-7)(x+7)\).

Step 2: Simplify the Expression

Next, simplify the expression by cancelling out common factors in the numerator and the denominator. As \(x-7\) is common to both the numerator and the denominator, the expression simplifies to: \(\frac{(x-7)}{(x+7)}\).

Step 3: Identify Excluded Values

Find any values for x that cause division by zero in the denominator. The domain of the function excludes these values because division by zero is undefined in mathematics. By setting x+7 equal to zero, we find that \(x=-7\) must be excluded from the domain of the simplified rational expression.