Divide using synthetic division $$ ( 6x^{4}+4x^{3}+4x^{2}+2x+5 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&6&4&4&2&5\\& & 24& 112& 464& \color{black}{1864} \\ \hline &\color{blue}{6}&\color{blue}{28}&\color{blue}{116}&\color{blue}{466}&\color{orangered}{1869} \end{array} $$The solution is:
$$ \frac{ 6x^{4}+4x^{3}+4x^{2}+2x+5 }{ x-4 } = \color{blue}{6x^{3}+28x^{2}+116x+466} ~+~ \frac{ \color{red}{ 1869 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&4&4&2&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 6 }&4&4&2&5\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&4&4&2&5\\& & \color{blue}{24} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 24 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrr}4&6&\color{orangered}{ 4 }&4&2&5\\& & \color{orangered}{24} & & & \\ \hline &6&\color{orangered}{28}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 28 } = \color{blue}{ 112 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&4&4&2&5\\& & 24& \color{blue}{112} & & \\ \hline &6&\color{blue}{28}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 112 } = \color{orangered}{ 116 } $
$$ \begin{array}{c|rrrrr}4&6&4&\color{orangered}{ 4 }&2&5\\& & 24& \color{orangered}{112} & & \\ \hline &6&28&\color{orangered}{116}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 116 } = \color{blue}{ 464 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&4&4&2&5\\& & 24& 112& \color{blue}{464} & \\ \hline &6&28&\color{blue}{116}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 464 } = \color{orangered}{ 466 } $
$$ \begin{array}{c|rrrrr}4&6&4&4&\color{orangered}{ 2 }&5\\& & 24& 112& \color{orangered}{464} & \\ \hline &6&28&116&\color{orangered}{466}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 466 } = \color{blue}{ 1864 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&4&4&2&5\\& & 24& 112& 464& \color{blue}{1864} \\ \hline &6&28&116&\color{blue}{466}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 1864 } = \color{orangered}{ 1869 } $
$$ \begin{array}{c|rrrrr}4&6&4&4&2&\color{orangered}{ 5 }\\& & 24& 112& 464& \color{orangered}{1864} \\ \hline &\color{blue}{6}&\color{blue}{28}&\color{blue}{116}&\color{blue}{466}&\color{orangered}{1869} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}+28x^{2}+116x+466 } $ with a remainder of $ \color{red}{ 1869 } $.