Divide using synthetic division $$ ( 2x^{3}-x^{2}-18x+32 ) \div ( 2x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&2&-1&-18&32\\& & 12& 66& \color{black}{288} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{48}&\color{orangered}{320} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-x^{2}-18x+32 }{ x-6 } = \color{blue}{2x^{2}+11x+48} ~+~ \frac{ \color{red}{ 320 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-1&-18&32\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 2 }&-1&-18&32\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 2 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-1&-18&32\\& & \color{blue}{12} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 12 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}6&2&\color{orangered}{ -1 }&-18&32\\& & \color{orangered}{12} & & \\ \hline &2&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 11 } = \color{blue}{ 66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-1&-18&32\\& & 12& \color{blue}{66} & \\ \hline &2&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 66 } = \color{orangered}{ 48 } $
$$ \begin{array}{c|rrrr}6&2&-1&\color{orangered}{ -18 }&32\\& & 12& \color{orangered}{66} & \\ \hline &2&11&\color{orangered}{48}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 48 } = \color{blue}{ 288 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-1&-18&32\\& & 12& 66& \color{blue}{288} \\ \hline &2&11&\color{blue}{48}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ 288 } = \color{orangered}{ 320 } $
$$ \begin{array}{c|rrrr}6&2&-1&-18&\color{orangered}{ 32 }\\& & 12& 66& \color{orangered}{288} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{48}&\color{orangered}{320} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+11x+48 } $ with a remainder of $ \color{red}{ 320 } $.