x3-10×2+32x-48x-6

Find where the expression x3-10×2+32x-48x-6 is undefined.

x=6

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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=3

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 | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 |

Divide the highest order term in the dividend x3 by the highest order term in divisor x.

x2 | |||||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 |

Multiply the new quotient term by the divisor.

x2 | |||||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

+ | x3 | – | 6×2 |

The expression needs to be subtracted from the dividend, so change all the signs in x3-6×2

x2 | |||||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 |

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

x2 | |||||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 |

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

x2 | |||||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x |

Divide the highest order term in the dividend -4×2 by the highest order term in divisor x.

x2 | – | 4x | |||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x |

Multiply the new quotient term by the divisor.

x2 | – | 4x | |||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

– | 4×2 | + | 24x |

The expression needs to be subtracted from the dividend, so change all the signs in -4×2+24x

x2 | – | 4x | |||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x |

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

x2 | – | 4x | |||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x |

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

x2 | – | 4x | |||||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x | – | 48 |

Divide the highest order term in the dividend 8x by the highest order term in divisor x.

x2 | – | 4x | + | 8 | |||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x | – | 48 |

Multiply the new quotient term by the divisor.

x2 | – | 4x | + | 8 | |||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x | – | 48 | ||||||||

+ | 8x | – | 48 |

The expression needs to be subtracted from the dividend, so change all the signs in 8x-48

x2 | – | 4x | + | 8 | |||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x | – | 48 | ||||||||

– | 8x | + | 48 |

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

x2 | – | 4x | + | 8 | |||||||

x | – | 6 | x3 | – | 10×2 | + | 32x | – | 48 | ||

– | x3 | + | 6×2 | ||||||||

– | 4×2 | + | 32x | ||||||||

+ | 4×2 | – | 24x | ||||||||

+ | 8x | – | 48 | ||||||||

– | 8x | + | 48 | ||||||||

0 |

Since the remander is 0, the final answer is the quotient.

x2-4x+8

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

y=x2-4x

y=x2-4x

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: y=x2-4x

Graph (x^3-10x^2+32x-48)/(x-6)