Divide using synthetic division $$ ( x^{4}-18x^{3}+95x^{2}+36x-194 ) \div ( 4x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-9&1&-18&95&36&-194\\& & -9& 243& -3042& \color{black}{27054} \\ \hline &\color{blue}{1}&\color{blue}{-27}&\color{blue}{338}&\color{blue}{-3006}&\color{orangered}{26860} \end{array} $$The solution is:
$$ \frac{ x^{4}-18x^{3}+95x^{2}+36x-194 }{ x+9 } = \color{blue}{x^{3}-27x^{2}+338x-3006} ~+~ \frac{ \color{red}{ 26860 } }{ x+9 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&-18&95&36&-194\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-9&\color{orangered}{ 1 }&-18&95&36&-194\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 1 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&-18&95&36&-194\\& & \color{blue}{-9} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrrr}-9&1&\color{orangered}{ -18 }&95&36&-194\\& & \color{orangered}{-9} & & & \\ \hline &1&\color{orangered}{-27}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 243 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&-18&95&36&-194\\& & -9& \color{blue}{243} & & \\ \hline &1&\color{blue}{-27}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 95 } + \color{orangered}{ 243 } = \color{orangered}{ 338 } $
$$ \begin{array}{c|rrrrr}-9&1&-18&\color{orangered}{ 95 }&36&-194\\& & -9& \color{orangered}{243} & & \\ \hline &1&-27&\color{orangered}{338}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 338 } = \color{blue}{ -3042 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&-18&95&36&-194\\& & -9& 243& \color{blue}{-3042} & \\ \hline &1&-27&\color{blue}{338}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -3042 \right) } = \color{orangered}{ -3006 } $
$$ \begin{array}{c|rrrrr}-9&1&-18&95&\color{orangered}{ 36 }&-194\\& & -9& 243& \color{orangered}{-3042} & \\ \hline &1&-27&338&\color{orangered}{-3006}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -3006 \right) } = \color{blue}{ 27054 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&-18&95&36&-194\\& & -9& 243& -3042& \color{blue}{27054} \\ \hline &1&-27&338&\color{blue}{-3006}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -194 } + \color{orangered}{ 27054 } = \color{orangered}{ 26860 } $
$$ \begin{array}{c|rrrrr}-9&1&-18&95&36&\color{orangered}{ -194 }\\& & -9& 243& -3042& \color{orangered}{27054} \\ \hline &\color{blue}{1}&\color{blue}{-27}&\color{blue}{338}&\color{blue}{-3006}&\color{orangered}{26860} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-27x^{2}+338x-3006 } $ with a remainder of $ \color{red}{ 26860 } $.