Divide using synthetic division $$ ( 8x^{3}+17x^{2}-30x-72 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&8&17&-30&-72\\& & 24& 123& \color{black}{279} \\ \hline &\color{blue}{8}&\color{blue}{41}&\color{blue}{93}&\color{orangered}{207} \end{array} $$The solution is:
$$ \frac{ 8x^{3}+17x^{2}-30x-72 }{ x-3 } = \color{blue}{8x^{2}+41x+93} ~+~ \frac{ \color{red}{ 207 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&8&17&-30&-72\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 8 }&17&-30&-72\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 8 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&8&17&-30&-72\\& & \color{blue}{24} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 24 } = \color{orangered}{ 41 } $
$$ \begin{array}{c|rrrr}3&8&\color{orangered}{ 17 }&-30&-72\\& & \color{orangered}{24} & & \\ \hline &8&\color{orangered}{41}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 41 } = \color{blue}{ 123 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&8&17&-30&-72\\& & 24& \color{blue}{123} & \\ \hline &8&\color{blue}{41}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 123 } = \color{orangered}{ 93 } $
$$ \begin{array}{c|rrrr}3&8&17&\color{orangered}{ -30 }&-72\\& & 24& \color{orangered}{123} & \\ \hline &8&41&\color{orangered}{93}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 93 } = \color{blue}{ 279 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&8&17&-30&-72\\& & 24& 123& \color{blue}{279} \\ \hline &8&41&\color{blue}{93}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -72 } + \color{orangered}{ 279 } = \color{orangered}{ 207 } $
$$ \begin{array}{c|rrrr}3&8&17&-30&\color{orangered}{ -72 }\\& & 24& 123& \color{orangered}{279} \\ \hline &\color{blue}{8}&\color{blue}{41}&\color{blue}{93}&\color{orangered}{207} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}+41x+93 } $ with a remainder of $ \color{red}{ 207 } $.