Divide using synthetic division $$ ( -x^{3}-18x^{2}+22x-5 ) \div ( 2x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&-1&-18&22&-5\\& & -5& -115& \color{black}{-465} \\ \hline &\color{blue}{-1}&\color{blue}{-23}&\color{blue}{-93}&\color{orangered}{-470} \end{array} $$The solution is:
$$ \frac{ -x^{3}-18x^{2}+22x-5 }{ x-5 } = \color{blue}{-x^{2}-23x-93} \color{red}{~-~} \frac{ \color{red}{ 470 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&-18&22&-5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ -1 }&-18&22&-5\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&-18&22&-5\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrr}5&-1&\color{orangered}{ -18 }&22&-5\\& & \color{orangered}{-5} & & \\ \hline &-1&\color{orangered}{-23}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ -115 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&-18&22&-5\\& & -5& \color{blue}{-115} & \\ \hline &-1&\color{blue}{-23}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 22 } + \color{orangered}{ \left( -115 \right) } = \color{orangered}{ -93 } $
$$ \begin{array}{c|rrrr}5&-1&-18&\color{orangered}{ 22 }&-5\\& & -5& \color{orangered}{-115} & \\ \hline &-1&-23&\color{orangered}{-93}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -93 \right) } = \color{blue}{ -465 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&-18&22&-5\\& & -5& -115& \color{blue}{-465} \\ \hline &-1&-23&\color{blue}{-93}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -465 \right) } = \color{orangered}{ -470 } $
$$ \begin{array}{c|rrrr}5&-1&-18&22&\color{orangered}{ -5 }\\& & -5& -115& \color{orangered}{-465} \\ \hline &\color{blue}{-1}&\color{blue}{-23}&\color{blue}{-93}&\color{orangered}{-470} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-23x-93 } $ with a remainder of $ \color{red}{ -470 } $.