Divide using synthetic division $$ ( 13x^{4}-2x^{3}+x^{2}-x+97 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&13&-2&1&-1&97\\& & -39& 123& -372& \color{black}{1119} \\ \hline &\color{blue}{13}&\color{blue}{-41}&\color{blue}{124}&\color{blue}{-373}&\color{orangered}{1216} \end{array} $$The solution is:
$$ \frac{ 13x^{4}-2x^{3}+x^{2}-x+97 }{ x+3 } = \color{blue}{13x^{3}-41x^{2}+124x-373} ~+~ \frac{ \color{red}{ 1216 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&13&-2&1&-1&97\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 13 }&-2&1&-1&97\\& & & & & \\ \hline &\color{orangered}{13}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 13 } = \color{blue}{ -39 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&13&-2&1&-1&97\\& & \color{blue}{-39} & & & \\ \hline &\color{blue}{13}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -39 \right) } = \color{orangered}{ -41 } $
$$ \begin{array}{c|rrrrr}-3&13&\color{orangered}{ -2 }&1&-1&97\\& & \color{orangered}{-39} & & & \\ \hline &13&\color{orangered}{-41}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -41 \right) } = \color{blue}{ 123 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&13&-2&1&-1&97\\& & -39& \color{blue}{123} & & \\ \hline &13&\color{blue}{-41}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 123 } = \color{orangered}{ 124 } $
$$ \begin{array}{c|rrrrr}-3&13&-2&\color{orangered}{ 1 }&-1&97\\& & -39& \color{orangered}{123} & & \\ \hline &13&-41&\color{orangered}{124}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 124 } = \color{blue}{ -372 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&13&-2&1&-1&97\\& & -39& 123& \color{blue}{-372} & \\ \hline &13&-41&\color{blue}{124}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -372 \right) } = \color{orangered}{ -373 } $
$$ \begin{array}{c|rrrrr}-3&13&-2&1&\color{orangered}{ -1 }&97\\& & -39& 123& \color{orangered}{-372} & \\ \hline &13&-41&124&\color{orangered}{-373}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -373 \right) } = \color{blue}{ 1119 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&13&-2&1&-1&97\\& & -39& 123& -372& \color{blue}{1119} \\ \hline &13&-41&124&\color{blue}{-373}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 97 } + \color{orangered}{ 1119 } = \color{orangered}{ 1216 } $
$$ \begin{array}{c|rrrrr}-3&13&-2&1&-1&\color{orangered}{ 97 }\\& & -39& 123& -372& \color{orangered}{1119} \\ \hline &\color{blue}{13}&\color{blue}{-41}&\color{blue}{124}&\color{blue}{-373}&\color{orangered}{1216} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 13x^{3}-41x^{2}+124x-373 } $ with a remainder of $ \color{red}{ 1216 } $.