identify the asymptotes. give your answers in exact form. do not round.\n\n q(x)=\frac{x^{3}+4 x^{2}-3…

identify the asymptotes. give your answers in exact form. do not round.\n\n q(x)=\frac{x^{3}+4 x^{2}-3 x+1}{x^{2}-5} \n\nseparate multiple equations of asymptotes with commas as necessary. select \none\ if applicable.\n\npart 1 of 2\n\nequation(s) of the vertical asymptote(s):\n\npart 2 of 2\n\nequation(s) of the horizontal asymptote(s):\n\nequation(s) of the slant asymptote(s):

identify the asymptotes. give your answers in exact form. do not round.\n\n q(x)=\frac{x^{3}+4 x^{2}-3 x+1}{x^{2}-5} \n\nseparate multiple equations of asymptotes with commas as necessary. select \none\ if applicable.\n\npart 1 of 2\n\nequation(s) of the vertical asymptote(s):\n\npart 2 of 2\n\nequation(s) of the horizontal asymptote(s):\n\nequation(s) of the slant asymptote(s):

Answer

Answer:

Part 1 of 2

Equation(s) of the vertical asymptote(s): (x = \sqrt{5},x=-\sqrt{5})

Part 2 of 2

Equation(s) of the horizontal asymptote(s): None Equation(s) of the slant asymptote(s): (y=x + 4)

Explanation:

Step1: Find vertical asymptotes

Set the denominator (x^{2}-5 = 0). Using the formula (a^{2}-b^{2}=(a + b)(a - b)), where (a=x) and (b=\sqrt{5}), we have ((x+\sqrt{5})(x - \sqrt{5})=0). Solving (x+\sqrt{5}=0) gives (x=-\sqrt{5}), and solving (x-\sqrt{5}=0) gives (x=\sqrt{5}).

Step2: Check for horizontal asymptote

The degree of the numerator (n = 3) (since the highest - power term in (x^{3}+4x^{2}-3x + 1) is (x^{3})) and the degree of the denominator (m=2) (since the highest - power term in (x^{2}-5) is (x^{2})). When (n>m) (here (n - m=3 - 2 = 1)), there is no horizontal asymptote.

Step3: Find slant asymptote

Use polynomial long - division. Divide (x^{3}+4x^{2}-3x + 1) by (x^{2}-5). [ \begin{align*} \frac{x^{3}+4x^{2}-3x + 1}{x^{2}-5}&=x + 4+\frac{2x+21}{x^{2}-5}\ \end{align*} ] As (x\to\pm\infty), (\frac{2x + 21}{x^{2}-5}\to0). So the slant asymptote is (y=x + 4).