Question

find all vertical asymptotes of the following function.
$f(x) = \frac{x - 4}{2x^2 + 34x + 140}$

Explanation:

Step1: Factor the denominator

First, factor out the common factor of 2 from the denominator: \(2x^{2}+34x + 140=2(x^{2}+17x + 70)\). Then factor the quadratic inside the parentheses: \(x^{2}+17x + 70=(x + 7)(x+10)\). So the denominator becomes \(2(x + 7)(x + 10)\).

Step2: Find values that make denominator zero

Set the denominator equal to zero: \(2(x + 7)(x + 10)=0\). Since 2 is never zero, we solve \((x + 7)(x + 10)=0\). Using the zero - product property, we get \(x+7 = 0\) or \(x + 10=0\), which gives \(x=-7\) or \(x=-10\).

Step3: Check if numerator is non - zero at these values

For \(x=-7\), the numerator is \(-7 - 4=-11
eq0\). For \(x=-10\), the numerator is \(-10 - 4=-14
eq0\). So these values make the function undefined and the numerator non - zero, so they are vertical asymptotes.

Answer:

The vertical asymptotes are \(x=-7\) and \(x = - 10\)