Divide using synthetic division $$ ( 6x^{3}+2x-3 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&6&0&2&-3\\& & 24& 96& \color{black}{392} \\ \hline &\color{blue}{6}&\color{blue}{24}&\color{blue}{98}&\color{orangered}{389} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+2x-3 }{ x-4 } = \color{blue}{6x^{2}+24x+98} ~+~ \frac{ \color{red}{ 389 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&0&2&-3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 6 }&0&2&-3\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&0&2&-3\\& & \color{blue}{24} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}4&6&\color{orangered}{ 0 }&2&-3\\& & \color{orangered}{24} & & \\ \hline &6&\color{orangered}{24}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 24 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&0&2&-3\\& & 24& \color{blue}{96} & \\ \hline &6&\color{blue}{24}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 96 } = \color{orangered}{ 98 } $
$$ \begin{array}{c|rrrr}4&6&0&\color{orangered}{ 2 }&-3\\& & 24& \color{orangered}{96} & \\ \hline &6&24&\color{orangered}{98}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 98 } = \color{blue}{ 392 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&0&2&-3\\& & 24& 96& \color{blue}{392} \\ \hline &6&24&\color{blue}{98}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 392 } = \color{orangered}{ 389 } $
$$ \begin{array}{c|rrrr}4&6&0&2&\color{orangered}{ -3 }\\& & 24& 96& \color{orangered}{392} \\ \hline &\color{blue}{6}&\color{blue}{24}&\color{blue}{98}&\color{orangered}{389} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+24x+98 } $ with a remainder of $ \color{red}{ 389 } $.