question 5 of 12, step 1 of 2 consider the following rational function. f(x) = (2x^3 + 1)/(x^2 - 16) step 1…

question 5 of 12, step 1 of 2 consider the following rational function. f(x) = (2x^3 + 1)/(x^2 - 16) step 1 of 2: find equations for the vertical asymptotes, if any, for the function. answer how to enter your answer (opens in new window) separate multiple equations with a comma. selecting a button will replace the entered answer value. the value of the button is used instead of the

question 5 of 12, step 1 of 2 consider the following rational function. f(x) = (2x^3 + 1)/(x^2 - 16) step 1 of 2: find equations for the vertical asymptotes, if any, for the function. answer how to enter your answer (opens in new window) separate multiple equations with a comma. selecting a button will replace the entered answer value. the value of the button is used instead of the

Answer

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non - zero. Set the denominator equal to zero: $x^{2}-16 = 0$.

Step2: Solve the equation for x

Factor the left - hand side using the difference of squares formula $a^{2}-b^{2}=(a + b)(a - b)$. Here, $a=x$ and $b = 4$, so $(x + 4)(x - 4)=0$. Then, by the zero - product property, $x+4=0$ or $x - 4=0$. Solving these equations gives $x=-4$ and $x = 4$.

Answer:

$x=-4,x = 4$