Divide using synthetic division $$ ( x^{3}-18x-42 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&0&-18&-42\\& & 5& 25& \color{black}{35} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{7}&\color{orangered}{-7} \end{array} $$The solution is:
$$ \frac{ x^{3}-18x-42 }{ x-5 } = \color{blue}{x^{2}+5x+7} \color{red}{~-~} \frac{ \color{red}{ 7 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-18&-42\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&0&-18&-42\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-18&-42\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ 0 }&-18&-42\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-18&-42\\& & 5& \color{blue}{25} & \\ \hline &1&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 25 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}5&1&0&\color{orangered}{ -18 }&-42\\& & 5& \color{orangered}{25} & \\ \hline &1&5&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 7 } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-18&-42\\& & 5& 25& \color{blue}{35} \\ \hline &1&5&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -42 } + \color{orangered}{ 35 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}5&1&0&-18&\color{orangered}{ -42 }\\& & 5& 25& \color{orangered}{35} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{7}&\color{orangered}{-7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+5x+7 } $ with a remainder of $ \color{red}{ -7 } $.