Question

choose whether the statement is true or false: the rational equation $y = \frac{2x^{2}+5x - 3}{x^{2}-1}$ has a slant asymptote. true; the degree of the numerator is greater than the degree of the denominator. false; the degree of the numerator is not one greater than the degree of the denominator

Explanation:

Step1: Determine degrees of numerator and denominator

The degree of the numerator $2x^{2}+5x - 3$ is 2 (highest - power of $x$ is 2), and the degree of the denominator $x^{2}-1$ is 2.

Step2: Recall slant - asymptote condition

A rational function $y=\frac{f(x)}{g(x)}$ has a slant asymptote when the degree of the numerator $f(x)$ is exactly one greater than the degree of the denominator $g(x)$. Here, the degrees are equal.

Answer:

False; the degree of the numerator is not one greater than the degree of the denominator