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Home > Algebra calculators > Division of two polynomials example
4. polynomial Long division example ( Enter your problem )
( Enter your problem )
  1. Examples
 		Other related methods
  1. Adding polynomials
  2. Subtracting polynomials
  3. Multiplying Polynomials
  4. polynomial Long division
  5. polynomial Synthetic division
  6. Remainder theorem
3. Multiplying Polynomials
(Previous method)		5. polynomial Synthetic division
(Next method)

1. Examples


1. Find division of `x^2-10x+25` and `x-5`

Solution:
Final Solution
 ```x``-``5`  
`color{blue}{x-5}``` `x^2``-` `10x``+` `25`  
 ``−`x^2``-`+`5x` `x xx (color{blue}{x-5})`
 `-` `5x``+` `25`  
 `-`+`5x``+`−`25` `color{green}{-5} xx (color{blue}{x-5})`
 `` `0`  

Final answer `= "Quotient" + (color{Magenta}{"Remainder"})/(color{blue}{"Divisor"})`.
`:.` Final answer = `x-5 + (color{Magenta}{0})/(color{blue}{x-5})`
 
Here, Divisor = `x-5`
Dividend = `x^2-10x+25`
Quotient = `x-5`
Remainder = `0`


Step by step division solution

Step - 1 :
1. Divide the first term of the dividend by the first term of the divisor : `(x^2)/(x)=color{green}{x}`

2. Write down the calculated result `color{green}{x}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{x} xx (color{blue}{x-5})=color{red}{x^2-5x}`

4. Subtract this result from the dividend
`(x^2-10x+25)-(color{red}{x^2-5x})=color{Magenta}{-5x+25}`

 ```x`  
`color{blue}{x-5}``` `x^2``-` `10x``+` `25`  
 ``−`x^2``-`+`5x` `color{green}{x} xx (color{blue}{x-5})`
 `-` `5x``+` `25`  


Step - 2 :
1. Divide the first term of the dividend by the first term of the divisor : `(-5x)/(x)=color{green}{-5}`

2. Write down the calculated result `color{green}{-5}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{-5} xx (color{blue}{x-5})=color{red}{-5x+25}`

4. Subtract this result from the remainder
`(-5x+25)-(color{red}{-5x+25})=color{Magenta}{0}`

 ```x``-``5`  
`color{blue}{x-5}``` `x^2``-` `10x``+` `25`  
 ``−`x^2``-`+`5x` `x xx (color{blue}{x-5})`
 `-` `5x``+` `25`  
 `-`+`5x``+`−`25` `color{green}{-5} xx (color{blue}{x-5})`
 `` `0`  


2. Find division of `x^3+4x^2-4x-16` and `x-2`

Solution:
Final Solution
 ```x^2``+``6x``+``8`  
`color{blue}{x-2}``` `x^3``+` `4x^2``-` `4x``-` `16`  
 ``−`x^3``-`+`2x^2` `x^2 xx (color{blue}{x-2})`
 `` `6x^2``-` `4x``-` `16`  
 ``−`6x^2``-`+`12x` `6x xx (color{blue}{x-2})`
 `` `8x``-` `16`  
 ``−`8x``-`+`16` `color{green}{8} xx (color{blue}{x-2})`
 `` `0`  

Final answer `= "Quotient" + (color{Magenta}{"Remainder"})/(color{blue}{"Divisor"})`.
`:.` Final answer = `x^2+6x+8 + (color{Magenta}{0})/(color{blue}{x-2})`
 
Here, Divisor = `x-2`
Dividend = `x^3+4x^2-4x-16`
Quotient = `x^2+6x+8`
Remainder = `0`


Step by step division solution

Step - 1 :
1. Divide the first term of the dividend by the first term of the divisor : `(x^3)/(x)=color{green}{x^2}`

2. Write down the calculated result `color{green}{x^2}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{x^2} xx (color{blue}{x-2})=color{red}{x^3-2x^2}`

4. Subtract this result from the dividend
`(x^3+4x^2-4x-16)-(color{red}{x^3-2x^2})=color{Magenta}{6x^2-4x-16}`

 ```x^2`  
`color{blue}{x-2}``` `x^3``+` `4x^2``-` `4x``-` `16`  
 ``−`x^3``-`+`2x^2` `color{green}{x^2} xx (color{blue}{x-2})`
 `` `6x^2``-` `4x``-` `16`  


Step - 2 :
1. Divide the first term of the dividend by the first term of the divisor : `(6x^2)/(x)=color{green}{6x}`

2. Write down the calculated result `color{green}{6x}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{6x} xx (color{blue}{x-2})=color{red}{6x^2-12x}`

4. Subtract this result from the remainder
`(6x^2-4x-16)-(color{red}{6x^2-12x})=color{Magenta}{8x-16}`

 ```x^2``+``6x`  
`color{blue}{x-2}``` `x^3``+` `4x^2``-` `4x``-` `16`  
 ``−`x^3``-`+`2x^2` `x^2 xx (color{blue}{x-2})`
 `` `6x^2``-` `4x``-` `16`  
 ``−`6x^2``-`+`12x` `color{green}{6x} xx (color{blue}{x-2})`
 `` `8x``-` `16`  


Step - 3 :
1. Divide the first term of the dividend by the first term of the divisor : `(8x)/(x)=color{green}{8}`

2. Write down the calculated result `color{green}{8}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{8} xx (color{blue}{x-2})=color{red}{8x-16}`

4. Subtract this result from the remainder
`(8x-16)-(color{red}{8x-16})=color{Magenta}{0}`

 ```x^2``+``6x``+``8`  
`color{blue}{x-2}``` `x^3``+` `4x^2``-` `4x``-` `16`  
 ``−`x^3``-`+`2x^2` `x^2 xx (color{blue}{x-2})`
 `` `6x^2``-` `4x``-` `16`  
 ``−`6x^2``-`+`12x` `6x xx (color{blue}{x-2})`
 `` `8x``-` `16`  
 ``−`8x``-`+`16` `color{green}{8} xx (color{blue}{x-2})`
 `` `0`  


3. Find division of `x^3-8y^3` and `x-2y`

Solution:
Final Solution
 `x^2+2xy+4y^2`  
`color{blue}{x-2y}``x^3-8y^3`  
 `x^3-2x^2y` `x^2 xx (color{blue}{x-2y})`
 `2x^2y-8y^3`  
 `2x^2y-4xy^2` `2xy xx (color{blue}{x-2y})`
 `4xy^2-8y^3`  
 `4xy^2-8y^3` `color{green}{4y^2} xx (color{blue}{x-2y})`
 `0`  

Final answer `= "Quotient" + (color{Magenta}{"Remainder"})/(color{blue}{"Divisor"})`.
`:.` Final answer = `x^2+2xy+4y^2 + (color{Magenta}{0})/(color{blue}{x-2y})`
 
Here, Divisor = `x-2y`
Dividend = `x^3-8y^3`
Quotient = `x^2+2xy+4y^2`
Remainder = `0`




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3. Multiplying Polynomials
(Previous method)		5. polynomial Synthetic division
(Next method)


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