Questions: Simplify the rational expression. Find all numbers that must be excluded from [ fracx^2+18 x+81x^2-81 ]

Solution Steps

To simplify the given rational expression, we need to factor both the numerator and the denominator. The numerator is a perfect square trinomial, and the denominator is a difference of squares. After factoring, we can simplify the expression by canceling out common factors. Additionally, we need to find the values of \( x \) that make the denominator zero, as these values must be excluded from the domain.

The given rational expression is

\[ \frac{x^{2} + 18x + 81}{x^{2} - 81}. \]

We factor the numerator \(x^{2} + 18x + 81\) as

\[ (x + 9)^{2} \]

and the denominator \(x^{2} - 81\) as

\[ (x - 9)(x + 9). \]

Substituting the factored forms into the expression, we have:

\[ \frac{(x + 9)^{2}}{(x - 9)(x + 9)}. \]

We can cancel the common factor \((x + 9)\) from the numerator and denominator, resulting in:

\[ \frac{x + 9}{x - 9}. \]

To find the values that must be excluded from the domain, we set the denominator equal to zero:

\[ x^{2} - 81 = 0. \]

Solving this gives us:

\[ x = 9 \quad \text{and} \quad x = -9. \]

Thus, the excluded values are \(x = 9\) and \(x = -9\).

Final Answer