Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x+27 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-27&1&17&70&-54\\& & -27& 270& \color{black}{-9180} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{340}&\color{orangered}{-9234} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x+27 } = \color{blue}{x^{2}-10x+340} \color{red}{~-~} \frac{ \color{red}{ 9234 } }{ x+27 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 27 = 0 $ ( $ x = \color{blue}{ -27 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-27}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-27&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -27 } \cdot \color{blue}{ 1 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-27}&1&17&70&-54\\& & \color{blue}{-27} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-27&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{-27} & & \\ \hline &1&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -27 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 270 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-27}&1&17&70&-54\\& & -27& \color{blue}{270} & \\ \hline &1&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 270 } = \color{orangered}{ 340 } $
$$ \begin{array}{c|rrrr}-27&1&17&\color{orangered}{ 70 }&-54\\& & -27& \color{orangered}{270} & \\ \hline &1&-10&\color{orangered}{340}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -27 } \cdot \color{blue}{ 340 } = \color{blue}{ -9180 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-27}&1&17&70&-54\\& & -27& 270& \color{blue}{-9180} \\ \hline &1&-10&\color{blue}{340}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ \left( -9180 \right) } = \color{orangered}{ -9234 } $
$$ \begin{array}{c|rrrr}-27&1&17&70&\color{orangered}{ -54 }\\& & -27& 270& \color{orangered}{-9180} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{340}&\color{orangered}{-9234} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-10x+340 } $ with a remainder of $ \color{red}{ -9234 } $.