Divide using synthetic division $$ ( 2x^{3}-10x^{2}-71x-9 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&2&-10&-71&-9\\& & 18& 72& \color{black}{9} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-10x^{2}-71x-9 }{ x-9 } = \color{blue}{2x^{2}+8x+1} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{9}&2&-10&-71&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 2 }&-10&-71&-9\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 2 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&2&-10&-71&-9\\& & \color{blue}{18} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 18 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}9&2&\color{orangered}{ -10 }&-71&-9\\& & \color{orangered}{18} & & \\ \hline &2&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 8 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&2&-10&-71&-9\\& & 18& \color{blue}{72} & \\ \hline &2&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -71 } + \color{orangered}{ 72 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}9&2&-10&\color{orangered}{ -71 }&-9\\& & 18& \color{orangered}{72} & \\ \hline &2&8&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&2&-10&-71&-9\\& & 18& 72& \color{blue}{9} \\ \hline &2&8&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 9 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}9&2&-10&-71&\color{orangered}{ -9 }\\& & 18& 72& \color{orangered}{9} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+8x+1 } $ with a remainder of $ \color{red}{ 0 } $.