Divide using synthetic division $$ ( 4x^{5}-3x^{3}+3x^{2}-2x+4 ) \div ( x-2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&4&0&-3&3&-2&4\\& & 8& 16& 26& 58& \color{black}{112} \\ \hline &\color{blue}{4}&\color{blue}{8}&\color{blue}{13}&\color{blue}{29}&\color{blue}{56}&\color{orangered}{116} \end{array} $$The solution is:
$$ \dfrac{ 4x^{5}-3x^{3}+3x^{2}-2x+4 }{ x-2 } = \color{blue}{4x^{4}+8x^{3}+13x^{2}+29x+56} ~+~ \dfrac{ \color{red}{ 116 } }{ x-2 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 4 }&0&-3&3&-2&4\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & \color{blue}{8} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}2&4&\color{orangered}{ 0 }&-3&3&-2&4\\& & \color{orangered}{8} & & & & \\ \hline &4&\color{orangered}{8}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & 8& \color{blue}{16} & & & \\ \hline &4&\color{blue}{8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 16 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}2&4&0&\color{orangered}{ -3 }&3&-2&4\\& & 8& \color{orangered}{16} & & & \\ \hline &4&8&\color{orangered}{13}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & 8& 16& \color{blue}{26} & & \\ \hline &4&8&\color{blue}{13}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 26 } = \color{orangered}{ 29 } $
$$ \begin{array}{c|rrrrrr}2&4&0&-3&\color{orangered}{ 3 }&-2&4\\& & 8& 16& \color{orangered}{26} & & \\ \hline &4&8&13&\color{orangered}{29}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 29 } = \color{blue}{ 58 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & 8& 16& 26& \color{blue}{58} & \\ \hline &4&8&13&\color{blue}{29}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 58 } = \color{orangered}{ 56 } $
$$ \begin{array}{c|rrrrrr}2&4&0&-3&3&\color{orangered}{ -2 }&4\\& & 8& 16& 26& \color{orangered}{58} & \\ \hline &4&8&13&29&\color{orangered}{56}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 56 } = \color{blue}{ 112 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&0&-3&3&-2&4\\& & 8& 16& 26& 58& \color{blue}{112} \\ \hline &4&8&13&29&\color{blue}{56}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 112 } = \color{orangered}{ 116 } $
$$ \begin{array}{c|rrrrrr}2&4&0&-3&3&-2&\color{orangered}{ 4 }\\& & 8& 16& 26& 58& \color{orangered}{112} \\ \hline &\color{blue}{4}&\color{blue}{8}&\color{blue}{13}&\color{blue}{29}&\color{blue}{56}&\color{orangered}{116} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+8x^{3}+13x^{2}+29x+56 } $ with a remainder of $ \color{red}{ 116 } $.