3×3-5×2-4x-82×2+x

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

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 |

Divide the highest order term in the dividend 3×3 by the highest order term in divisor 2×2.

3×2 | |||||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 |

Multiply the new quotient term by the divisor.

3×2 | |||||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

+ | 3×3 | + | 3×22 | + | 0 |

The expression needs to be subtracted from the dividend, so change all the signs in 3×3+3×22+0

3×2 | |||||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 |

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

3×2 | |||||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x |

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

3×2 | |||||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x | – | 8 |

Divide the highest order term in the dividend -13×22 by the highest order term in divisor 2×2.

3×2 | – | 134 | |||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x | – | 8 |

Multiply the new quotient term by the divisor.

3×2 | – | 134 | |||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x | – | 8 | ||||||||

– | 13×22 | – | 13×4 | + | 0 |

The expression needs to be subtracted from the dividend, so change all the signs in -13×22-13×4+0

3×2 | – | 134 | |||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x | – | 8 | ||||||||

+ | 13×22 | + | 13×4 | – | 0 |

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

3×2 | – | 134 | |||||||||||

2×2 | + | x | + | 0 | 3×3 | – | 5×2 | – | 4x | – | 8 | ||

– | 3×3 | – | 3×22 | – | 0 | ||||||||

– | 13×22 | – | 4x | – | 8 | ||||||||

+ | 13×22 | + | 13×4 | – | 0 | ||||||||

– | 3×4 | – | 8 |

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

3×2-134+-3×4-82×2+x

Divide (3x^3-5x^2-4x-8)/(2x^2+x)