4x^3+5x^2+6/(x^2-3x-4) How to find VERTICAL AND HORIZONTALasymptotes for graph of the rational funtion F(x)?

Answer 1

Find the asymptotes.

I assume you mean.

(4x³ + 5x² + 6) / (x² - 3x - 4)

= (4x³ + 5x² + 6) / [(x + 1)(x - 4)]

The vertical asymptotes can be found when the denominator equals zero. The vertical asymptotes are:

x = -1, 4
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The numerator is a polynomial of higher degree than the denominator, so there is no horizontal asymptote.
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The numerator is a polynomial of one degree higher than the denominator, so there is a slant asymptote.

Using long division we have:

(4x³ + 5x² + 6) / (x² - 3x - 4)

= (4x + 17) + remainder

The slant asymptote is:

y = 4x + 17