Question

standard 9 homework
score: 7/27 answered: 7/20
question 8
find the vertical asymptotes and holes (removable discontinuities) of the rational function.
$f(x)=\frac{x - 10}{(x - 7)(x + 8)}$
the holes are at $x=$
the vertical asymptotes are $x=$
if there is more than one of each, enter your answers separated by a comma. if there is not a hole or
asymptote, enter dne.
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Explanation:

Step1: Analyze holes

Holes occur when a factor cancels in numerator and denominator. Here, numerator \(x - 10\) and denominator \((x - 7)(x + 8)\) have no common factors. So, holes: \(x=\text{DNE}\)

Step2: Analyze vertical asymptotes

Vertical asymptotes occur where denominator is zero (after canceling common factors). Set \((x - 7)(x + 8)=0\). Using zero - product property \(x-7 = 0\) gives \(x = 7\) and \(x+8=0\) gives \(x=-8\)

Answer:

The holes are at \(x=\text{DNE}\).
The vertical asymptotes are \(x = 7,-8\)