Questions: a. Find the slant asymptote of the graph of the rational function and b. Use the slant asymptote to graph the rational function. f(x) = (x^3 - 27) / (x^2 - 25)

Solution Steps

To find the slant asymptote of the rational function \( f(x) = \frac{x^3 - 27}{x^2 - 25} \), we perform polynomial long division of the numerator by the denominator.

1. Divide \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 - 25 \) to get \( x^3 - 25x \).
3. Subtract \( x^3 - 25x \) from \( x^3 - 27 \) to get \( 25x - 27 \).
4. Divide \( 25x \) by \( x^2 \) to get \( 0 \) (since the degree of the remainder is less than the degree of the divisor).

The quotient is \( x \), which means the slant asymptote is \( y = x \).

Final Answer