Question

question
consider the graph of the function ( f(x)=\frac{x^{3}+7x}{x^{3}-5x^{2}-14x} ).
what are the vertical asymptotes? enter only the ( x ) values separated by a comma.
provide your answer below:
( x=square )

Explanation:

Step1: Factor numerator and denominator

Factor \(x^{3}+7x=x(x^{2} + 7)\) and \(x^{3}-5x^{2}-14x=x(x^{2}-5x - 14)=x(x - 7)(x+2)\). So \(f(x)=\frac{x(x^{2}+7)}{x(x - 7)(x + 2)}\).

Step2: Simplify the function

Cancel out the common factor \(x\) (note \(x
eq0\)), get \(f(x)=\frac{x^{2}+7}{(x - 7)(x + 2)}\).

Step3: Find vertical asymptotes

Set the denominator \((x - 7)(x + 2)=0\). Solve \(x-7 = 0\) gives \(x = 7\), solve \(x+2=0\) gives \(x=-2\).

Answer:

\(-2,7\)