Divide using synthetic division $$ ( 5x^{3}-17x^{2}-17x+67 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&5&-17&-17&67\\& & 15& -6& \color{black}{-69} \\ \hline &\color{blue}{5}&\color{blue}{-2}&\color{blue}{-23}&\color{orangered}{-2} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-17x^{2}-17x+67 }{ x-3 } = \color{blue}{5x^{2}-2x-23} \color{red}{~-~} \frac{ \color{red}{ 2 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&5&-17&-17&67\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 5 }&-17&-17&67\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 5 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&5&-17&-17&67\\& & \color{blue}{15} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 15 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}3&5&\color{orangered}{ -17 }&-17&67\\& & \color{orangered}{15} & & \\ \hline &5&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&5&-17&-17&67\\& & 15& \color{blue}{-6} & \\ \hline &5&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrr}3&5&-17&\color{orangered}{ -17 }&67\\& & 15& \color{orangered}{-6} & \\ \hline &5&-2&\color{orangered}{-23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ -69 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&5&-17&-17&67\\& & 15& -6& \color{blue}{-69} \\ \hline &5&-2&\color{blue}{-23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 67 } + \color{orangered}{ \left( -69 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}3&5&-17&-17&\color{orangered}{ 67 }\\& & 15& -6& \color{orangered}{-69} \\ \hline &\color{blue}{5}&\color{blue}{-2}&\color{blue}{-23}&\color{orangered}{-2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-2x-23 } $ with a remainder of $ \color{red}{ -2 } $.