Question

find all vertical asymptotes of the following function.

f(x)=\frac{10 x + 4}{x^{2}-64}

Explanation:

Step1: Factor the denominator

The denominator \(x^{2}-64\) can be factored using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\). Here \(a=x\) and \(b = 8\), so \(x^{2}-64=(x + 8)(x - 8)\).

Step2: Set the denominator equal to zero

To find the vertical asymptotes, we set the denominator equal to zero and solve for \(x\).
\((x + 8)(x - 8)=0\)
Using the zero - product property \(ab = 0\) implies \(a=0\) or \(b = 0\).
If \(x+8=0\), then \(x=-8\).
If \(x - 8=0\), then \(x = 8\).

Step3: Check for common factors with the numerator

The numerator is \(10x + 4=2(5x + 2)\). There are no common factors between the numerator \(10x + 4\) and the denominator \((x + 8)(x - 8)\).

Answer:

The vertical asymptotes are \(x=-8\) and \(x = 8\).