Divide using synthetic division $$ ( 2x^{3}-20x^{2}+56x-46 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&2&-20&56&-46\\& & 12& -48& \color{black}{48} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-20x^{2}+56x-46 }{ x-6 } = \color{blue}{2x^{2}-8x+8} ~+~ \frac{ \color{red}{ 2 } }{ x-6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-20&56&-46\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 2 }&-20&56&-46\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-20&56&-46\\& & \color{blue}{12} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 12 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}6&2&\color{orangered}{ -20 }&56&-46\\& & \color{orangered}{12} & & \\ \hline &2&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-20&56&-46\\& & 12& \color{blue}{-48} & \\ \hline &2&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 56 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}6&2&-20&\color{orangered}{ 56 }&-46\\& & 12& \color{orangered}{-48} & \\ \hline &2&-8&\color{orangered}{8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 8 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&2&-20&56&-46\\& & 12& -48& \color{blue}{48} \\ \hline &2&-8&\color{blue}{8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -46 } + \color{orangered}{ 48 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}6&2&-20&56&\color{orangered}{ -46 }\\& & 12& -48& \color{orangered}{48} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-8x+8 } $ with a remainder of $ \color{red}{ 2 } $.