Divide using synthetic division $$ ( x^{4}-3x^{3}-10x^{2}+42x-36 ) \div ( 3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&1&-3&-10&42&-36\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-10}&\color{blue}{42}&\color{orangered}{-36} \end{array} $$The solution is:
$$ \frac{ x^{4}-3x^{3}-10x^{2}+42x-36 }{ x } = \color{blue}{x^{3}-3x^{2}-10x+42} \color{red}{~-~} \frac{ \color{red}{ 36 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-3&-10&42&-36\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&-3&-10&42&-36\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-3&-10&42&-36\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ -3 }&-10&42&-36\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{-3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-3&-10&42&-36\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}0&1&-3&\color{orangered}{ -10 }&42&-36\\& & 0& \color{orangered}{0} & & \\ \hline &1&-3&\color{orangered}{-10}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-3&-10&42&-36\\& & 0& 0& \color{blue}{0} & \\ \hline &1&-3&\color{blue}{-10}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 42 } + \color{orangered}{ 0 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrrr}0&1&-3&-10&\color{orangered}{ 42 }&-36\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&-3&-10&\color{orangered}{42}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 42 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-3&-10&42&-36\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&-3&-10&\color{blue}{42}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 0 } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrrr}0&1&-3&-10&42&\color{orangered}{ -36 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-10}&\color{blue}{42}&\color{orangered}{-36} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-3x^{2}-10x+42 } $ with a remainder of $ \color{red}{ -36 } $.