Divide using synthetic division $$ ( x^{3}-13x^{2}+38x+10 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&-13&38&10\\& & 5& -40& \color{black}{-10} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-13x^{2}+38x+10 }{ x-5 } = \color{blue}{x^{2}-8x-2} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-13&38&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&-13&38&10\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-13&38&10\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 5 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ -13 }&38&10\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-13&38&10\\& & 5& \color{blue}{-40} & \\ \hline &1&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 38 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}5&1&-13&\color{orangered}{ 38 }&10\\& & 5& \color{orangered}{-40} & \\ \hline &1&-8&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-13&38&10\\& & 5& -40& \color{blue}{-10} \\ \hline &1&-8&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}5&1&-13&38&\color{orangered}{ 10 }\\& & 5& -40& \color{orangered}{-10} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-8x-2 } $ with a remainder of $ \color{red}{ 0 } $.