x2+11x+29x+7

Find where the expression x2+11x+29x+7 is undefined.

x=-7

Consider the rational function R(x)=axnbxm where n is the degree of the numerator and m is the degree of the denominator.

1. If n<m, then the x-axis, y=0, is the horizontal asymptote.

2. If n=m, then the horizontal asymptote is the line y=ab.

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

Find n and m.

n=2

m=1

Since n>m, there is no horizontal asymptote.

No Horizontal Asymptotes

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of 0.

x | + | 7 | x2 | + | 11x | + | 29 |

Divide the highest order term in the dividend x2 by the highest order term in divisor x.

x | |||||||||

x | + | 7 | x2 | + | 11x | + | 29 |

Multiply the new quotient term by the divisor.

x | |||||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

+ | x2 | + | 7x |

The expression needs to be subtracted from the dividend, so change all the signs in x2+7x

x | |||||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x |

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

x | |||||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x |

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

x | |||||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x | + | 29 |

Divide the highest order term in the dividend 4x by the highest order term in divisor x.

x | + | 4 | |||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x | + | 29 |

Multiply the new quotient term by the divisor.

x | + | 4 | |||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x | + | 29 | ||||||

+ | 4x | + | 28 |

The expression needs to be subtracted from the dividend, so change all the signs in 4x+28

x | + | 4 | |||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x | + | 29 | ||||||

– | 4x | – | 28 |

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

x | + | 4 | |||||||

x | + | 7 | x2 | + | 11x | + | 29 | ||

– | x2 | – | 7x | ||||||

+ | 4x | + | 29 | ||||||

– | 4x | – | 28 | ||||||

+ | 1 |

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

x+4+1x+7

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

y=x+4

y=x+4

This is the set of all asymptotes.

Vertical Asymptotes: x=-7

No Horizontal Asymptotes

Oblique Asymptotes: y=x+4

Graph (x^2+11x+29)/(x+7)