Question

find the vertical asymptotes of the graph
$f(x)=\frac{2(x + 2)^2(x - 3)}{3x^2(x + 3)(x - 5)}$

Explanation:

Step1: Find the denominator's zeros

Set the denominator \(3x^{2}(x + 3)(x-5)=0\).
Using the zero - product property \(ab = 0\Rightarrow a=0\) or \(b = 0\).
For \(3x^{2}=0\), we get \(x = 0\) (with multiplicity \(2\)).
For \(x + 3=0\), we get \(x=-3\).
For \(x - 5=0\), we get \(x = 5\).

Step2: Check for common factors with the numerator

The numerator is \(2(x + 2)^{2}(x-3)\).
There are no common factors between the numerator \(2(x + 2)^{2}(x - 3)\) and the denominator \(3x^{2}(x + 3)(x-5)\).

Answer:

The vertical asymptotes are \(x=-3\), \(x = 0\), and \(x = 5\).