Questions: Polynomial functions have [ Select ] vertical asymptotes.

Solution Steps

To determine the number of vertical asymptotes of a polynomial function, we need to understand the nature of polynomial functions and their denominators. Polynomial functions themselves do not have vertical asymptotes unless they are rational functions (ratios of polynomials). Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non-zero.

Vertical asymptotes occur in rational functions where the denominator is zero and the numerator is non-zero. For a rational function \( \frac{P(x)}{Q(x)} \), vertical asymptotes are found by solving \( Q(x) = 0 \) and ensuring \( P(x) \neq 0 \) at those points.

Given the numerator polynomial \( P(x) = 1x^2 + 2x + 3 \) and the denominator polynomial \( Q(x) = 0x^2 + 1x + 2 \), we need to find the roots of \( Q(x) \).

The denominator polynomial \( Q(x) = x + 2 \) has a root at \( x = -2 \). This is where the denominator becomes zero.

We need to check if the numerator \( P(x) \) is non-zero at \( x = -2 \): \[ P(-2) = 1(-2)^2 + 2(-2) + 3 = 4 - 4 + 3 = 3 \] Since \( P(-2) = 3 \neq 0 \), there is a vertical asymptote at \( x = -2 \).

Final Answer