3×5+14×3+13×2-12x+13×2+1

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

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 |

Divide the highest order term in the dividend 3×5 by the highest order term in divisor x2.

3×3 | |||||||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 |

Multiply the new quotient term by the divisor.

3×3 | |||||||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

+ | 3×5 | + | 0 | + | 3×3 |

The expression needs to be subtracted from the dividend, so change all the signs in 3×5+0+3×3

3×3 | |||||||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 |

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

3×3 | |||||||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 |

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

3×3 | |||||||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x |

Divide the highest order term in the dividend 11×3 by the highest order term in divisor x2.

3×3 | + | 0x2 | + | 11x | |||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x |

Multiply the new quotient term by the divisor.

3×3 | + | 0x2 | + | 11x | |||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

+ | 11×3 | + | 0 | + | 11x |

The expression needs to be subtracted from the dividend, so change all the signs in 11×3+0+11x

3×3 | + | 0x2 | + | 11x | |||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x |

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

3×3 | + | 0x2 | + | 11x | |||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x |

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

3×3 | + | 0x2 | + | 11x | |||||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x | + | 13 |

Divide the highest order term in the dividend 13×2 by the highest order term in divisor x2.

3×3 | + | 0x2 | + | 11x | + | 13 | |||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x | + | 13 |

Multiply the new quotient term by the divisor.

3×3 | + | 0x2 | + | 11x | + | 13 | |||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x | + | 13 | ||||||||||||

+ | 13×2 | + | 0 | + | 13 |

The expression needs to be subtracted from the dividend, so change all the signs in 13×2+0+13

3×3 | + | 0x2 | + | 11x | + | 13 | |||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x | + | 13 | ||||||||||||

– | 13×2 | – | 0 | – | 13 |

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

3×3 | + | 0x2 | + | 11x | + | 13 | |||||||||||

x2 | + | 0x | + | 1 | 3×5 | + | 0x4 | + | 14×3 | + | 13×2 | – | 12x | + | 13 | ||

– | 3×5 | – | 0 | – | 3×3 | ||||||||||||

+ | 11×3 | + | 13×2 | – | 12x | ||||||||||||

– | 11×3 | – | 0 | – | 11x | ||||||||||||

+ | 13×2 | – | 23x | + | 13 | ||||||||||||

– | 13×2 | – | 0 | – | 13 | ||||||||||||

– | 23x | + | 0 |

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

3×3+11x+13+-23xx2+1

Divide (3x^5+14x^3+13x^2-12x+13)/(x^2+1)