Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&9&-9&-58&5&24\\& & 18& 18& -80& \color{black}{-150} \\ \hline &\color{blue}{9}&\color{blue}{9}&\color{blue}{-40}&\color{blue}{-75}&\color{orangered}{-126} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x-2 } = \color{blue}{9x^{3}+9x^{2}-40x-75} \color{red}{~-~} \frac{ \color{red}{ 126 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&-9&-58&5&24\\& & \color{blue}{18} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 18 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}2&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{18} & & & \\ \hline &9&\color{orangered}{9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&-9&-58&5&24\\& & 18& \color{blue}{18} & & \\ \hline &9&\color{blue}{9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 18 } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrrr}2&9&-9&\color{orangered}{ -58 }&5&24\\& & 18& \color{orangered}{18} & & \\ \hline &9&9&\color{orangered}{-40}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -40 \right) } = \color{blue}{ -80 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&-9&-58&5&24\\& & 18& 18& \color{blue}{-80} & \\ \hline &9&9&\color{blue}{-40}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -80 \right) } = \color{orangered}{ -75 } $
$$ \begin{array}{c|rrrrr}2&9&-9&-58&\color{orangered}{ 5 }&24\\& & 18& 18& \color{orangered}{-80} & \\ \hline &9&9&-40&\color{orangered}{-75}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -75 \right) } = \color{blue}{ -150 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&-9&-58&5&24\\& & 18& 18& -80& \color{blue}{-150} \\ \hline &9&9&-40&\color{blue}{-75}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -150 \right) } = \color{orangered}{ -126 } $
$$ \begin{array}{c|rrrrr}2&9&-9&-58&5&\color{orangered}{ 24 }\\& & 18& 18& -80& \color{orangered}{-150} \\ \hline &\color{blue}{9}&\color{blue}{9}&\color{blue}{-40}&\color{blue}{-75}&\color{orangered}{-126} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+9x^{2}-40x-75 } $ with a remainder of $ \color{red}{ -126 } $.