Divide using synthetic division $$ ( x^{3}-2x^{2}-21x+35 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&-2&-21&35\\& & 5& 15& \color{black}{-30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-6}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ x^{3}-2x^{2}-21x+35 }{ x-5 } = \color{blue}{x^{2}+3x-6} ~+~ \frac{ \color{red}{ 5 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-2&-21&35\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&-2&-21&35\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-2&-21&35\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 5 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ -2 }&-21&35\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-2&-21&35\\& & 5& \color{blue}{15} & \\ \hline &1&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 15 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}5&1&-2&\color{orangered}{ -21 }&35\\& & 5& \color{orangered}{15} & \\ \hline &1&3&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-2&-21&35\\& & 5& 15& \color{blue}{-30} \\ \hline &1&3&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}5&1&-2&-21&\color{orangered}{ 35 }\\& & 5& 15& \color{orangered}{-30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-6}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+3x-6 } $ with a remainder of $ \color{red}{ 5 } $.