Divide using synthetic division $$ ( 3x^{3}-13x^{2}+7x-2 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&3&-13&7&-2\\& & 9& -12& \color{black}{-15} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{-5}&\color{orangered}{-17} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-13x^{2}+7x-2 }{ x-3 } = \color{blue}{3x^{2}-4x-5} \color{red}{~-~} \frac{ \color{red}{ 17 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-13&7&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&-13&7&-2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-13&7&-2\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 9 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ -13 }&7&-2\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-13&7&-2\\& & 9& \color{blue}{-12} & \\ \hline &3&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}3&3&-13&\color{orangered}{ 7 }&-2\\& & 9& \color{orangered}{-12} & \\ \hline &3&-4&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-13&7&-2\\& & 9& -12& \color{blue}{-15} \\ \hline &3&-4&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}3&3&-13&7&\color{orangered}{ -2 }\\& & 9& -12& \color{orangered}{-15} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{-5}&\color{orangered}{-17} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-4x-5 } $ with a remainder of $ \color{red}{ -17 } $.