Find where the expression is undefined.

Since as from the left and as from the right, then is a vertical asymptote.

Since as from the left and as from the right, then is a vertical asymptote.

List all of the vertical asymptotes:

Consider the rational function where is the degree of the numerator and is the degree of the denominator.

1. If , then the x-axis, , is the horizontal asymptote.

2. If , then the horizontal asymptote is the line .

3. If , then there is no horizontal asymptote (there is an oblique asymptote).

Find and .

Since , there is no horizontal asymptote.

No Horizontal Asymptotes

Factor using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

Expand .

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Reorder and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Subtract from .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .

+ | – | + | + | + | + |

Divide the highest order term in the dividend by the highest order term in divisor .

+ | – | + | + | + | + |

Multiply the new quotient term by the divisor.

+ | – | + | + | + | + | ||||||||||

+ | + | – |

The expression needs to be subtracted from the dividend, so change all the signs in

+ | – | + | + | + | + | ||||||||||

– | – | + |

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + |

Pull the next terms from the original dividend down into the current dividend.

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + |

Divide the highest order term in the dividend by the highest order term in divisor .

– | |||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + |

Multiply the new quotient term by the divisor.

– | |||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

– | – | + |

The expression needs to be subtracted from the dividend, so change all the signs in

– | |||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – |

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

– | |||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – |

Pull the next terms from the original dividend down into the current dividend.

– | |||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – | + |

Divide the highest order term in the dividend by the highest order term in divisor .

– | + | ||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – | + |

Multiply the new quotient term by the divisor.

– | + | ||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – | + | |||||||||||||

+ | + | – |

The expression needs to be subtracted from the dividend, so change all the signs in

– | + | ||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – | + | |||||||||||||

– | – | + |

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

– | + | ||||||||||||||

+ | – | + | + | + | + | ||||||||||

– | – | + | |||||||||||||

– | + | + | |||||||||||||

+ | + | – | |||||||||||||

+ | – | + | |||||||||||||

– | – | + | |||||||||||||

– | + |

The final answer is the quotient plus the remainder over the divisor.

The oblique asymptote is the polynomial portion of the long division result.

This is the set of all asymptotes.

Vertical Asymptotes:

No Horizontal Asymptotes

Oblique Asymptotes:

Find the Asymptotes f(x)=(x^4+1)/(x^2+5x-14)