determine the vertical asymptotes of the graph of the function. give your answers as equations in exact…

determine the vertical asymptotes of the graph of the function. give your answers as equations in exact form.\n\n$$ k ( t ) = \frac { t ^ { 4 } - 1 } { 3 t ^ { 2 } + 8 t - 1 } $$\n\nseparate multiple equations with commas as necessary. select \none\ if applicable.\n\nequation(s) of the vertical asymptote(s):

determine the vertical asymptotes of the graph of the function. give your answers as equations in exact form.\n\n$$ k ( t ) = \frac { t ^ { 4 } - 1 } { 3 t ^ { 2 } + 8 t - 1 } $$\n\nseparate multiple equations with commas as necessary. select \none\ if applicable.\n\nequation(s) of the vertical asymptote(s):

Answer

Answer:

$t=\frac{-4+\sqrt{19}}{3},t=\frac{-4-\sqrt{19}}{3}$

Explanation:

Step1: Find the denominator of the rational function

The denominator of the function (k(t)=\frac{t^{4}-1}{3t^{2}+8t - 1}) is (y = 3t^{2}+8t - 1).

Step2: Use the quadratic formula

For a quadratic equation (ax^{2}+bx + c=0) ((a = 3), (b = 8), (c=-1)), the quadratic formula is (t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}). Substitute the values: (t=\frac{-8\pm\sqrt{8^{2}-4\times3\times(-1)}}{2\times3}=\frac{-8\pm\sqrt{64 + 12}}{6}=\frac{-8\pm\sqrt{76}}{6}=\frac{-8\pm2\sqrt{19}}{6}=\frac{-4\pm\sqrt{19}}{3}).

Step3: Write the equations of vertical asymptotes

The vertical asymptotes of a rational function (y=\frac{f(t)}{g(t)}) occur at the values of (t) that make (g(t)=0) (provided the numerator (f(t)\neq0) at those values). Since (t^{4}-1\neq0) when (3t^{2}+8t - 1 = 0), the equations of the vertical asymptotes are (t=\frac{-4+\sqrt{19}}{3}) and (t=\frac{-4-\sqrt{19}}{3}).