How do you find all the asymptotes for function $f (x) = \frac{11 {(x - 5)}^{5} {(x + 6)}^{5}}{x^{10} + 11 x^{5} + 30}$ $f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)$ ?

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How do you find all the asymptotes for function $f (x) = \frac{11 {(x - 5)}^{5} {(x + 6)}^{5}}{x^{10} + 11 x^{5} + 30}$ $f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)$ ?

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Answer 1 · 2016-02-26T10:54:18

$x = - \sqrt[5]{6}$ $x=-root(5)6$ and $x = - \sqrt[5]{5}$ $x=-root(5)5$ are two vertical asymptotes. Horizontal asymptote is $y = 11$ $y=11$ .

Explanation:

To find asymptotes of $f (x) = \frac{11 {(x - 5)}^{5} \cdot {(x + 6)}^{5}}{x^{10} + 11 x^{5} + 30}$ $f(x)=(11(x−5)^5*(x+6)^5)/(x^10+11x^5+30)$ , we should factorize denominator $(x^{10} + 11 x^{5} + 30)$ $(x^10+11x^5+30)$

The factors of $(x^{10} + 11 x^{5} + 30)$ $(x^10+11x^5+30)$ are $(x^{5} + 5)$ $(x^5+5)$ and $(x^{5} + 6)$ $(x^5+6)$

As $(x + a)$ $(x+a)$ is a factor of $(x^{5} + a^{5})$ $(x^5+a^5)$ (as $- a$ $-a$ is zero of latter)

two factors of $(x^{10} + 11 x^{5} + 30)$ $(x^10+11x^5+30)$ are $x = - \sqrt[5]{6}$ $x=-root(5)6$ and $x = - \sqrt[5]{5}$ $x=-root(5)5$

and hence $x = - \sqrt[5]{6}$ $x=-root(5)6$ and $x = - \sqrt[5]{5}$ $x=-root(5)5$ are two vertical asymptotes.

It is also apparent that the highest degree of numerator is $11 x^{10}$ $11x^10$ and that of denominator is $x^{10}$ $x^10$ . As their ratio is $11$ $11$

Horizontal asymptote is $y = 11$ $y=11$ .

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How do you find all the asymptotes for function $f (x) = \frac{11 {(x - 5)}^{5} {(x + 6)}^{5}}{x^{10} + 11 x^{5} + 30}$ $f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)$ ?

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How do you find all the asymptotes for function f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)f(x)=11(x−5)5(x+6)5x10+11x5+30f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)?

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Explanation:

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How do you find all the asymptotes for function $f (x) = \frac{11 {(x - 5)}^{5} {(x + 6)}^{5}}{x^{10} + 11 x^{5} + 30}$ $f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)$ ?