Divide using synthetic division $$ ( 9x^{3}-83x^{2}-43x+26 ) \div ( 9x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&9&-83&-43&26\\& & -63& 1022& \color{black}{-6853} \\ \hline &\color{blue}{9}&\color{blue}{-146}&\color{blue}{979}&\color{orangered}{-6827} \end{array} $$The solution is:
$$ \frac{ 9x^{3}-83x^{2}-43x+26 }{ x+7 } = \color{blue}{9x^{2}-146x+979} \color{red}{~-~} \frac{ \color{red}{ 6827 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&9&-83&-43&26\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 9 }&-83&-43&26\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 9 } = \color{blue}{ -63 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&9&-83&-43&26\\& & \color{blue}{-63} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -83 } + \color{orangered}{ \left( -63 \right) } = \color{orangered}{ -146 } $
$$ \begin{array}{c|rrrr}-7&9&\color{orangered}{ -83 }&-43&26\\& & \color{orangered}{-63} & & \\ \hline &9&\color{orangered}{-146}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -146 \right) } = \color{blue}{ 1022 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&9&-83&-43&26\\& & -63& \color{blue}{1022} & \\ \hline &9&\color{blue}{-146}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -43 } + \color{orangered}{ 1022 } = \color{orangered}{ 979 } $
$$ \begin{array}{c|rrrr}-7&9&-83&\color{orangered}{ -43 }&26\\& & -63& \color{orangered}{1022} & \\ \hline &9&-146&\color{orangered}{979}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 979 } = \color{blue}{ -6853 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&9&-83&-43&26\\& & -63& 1022& \color{blue}{-6853} \\ \hline &9&-146&\color{blue}{979}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -6853 \right) } = \color{orangered}{ -6827 } $
$$ \begin{array}{c|rrrr}-7&9&-83&-43&\color{orangered}{ 26 }\\& & -63& 1022& \color{orangered}{-6853} \\ \hline &\color{blue}{9}&\color{blue}{-146}&\color{blue}{979}&\color{orangered}{-6827} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}-146x+979 } $ with a remainder of $ \color{red}{ -6827 } $.