Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+58x-24 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&2&-7&-17&58&-24\\& & 16& 72& 440& \color{black}{3984} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{55}&\color{blue}{498}&\color{orangered}{3960} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+58x-24 }{ x-8 } = \color{blue}{2x^{3}+9x^{2}+55x+498} ~+~ \frac{ \color{red}{ 3960 } }{ x-8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&2&-7&-17&58&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 2 }&-7&-17&58&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 2 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&2&-7&-17&58&-24\\& & \color{blue}{16} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 16 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}8&2&\color{orangered}{ -7 }&-17&58&-24\\& & \color{orangered}{16} & & & \\ \hline &2&\color{orangered}{9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 9 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&2&-7&-17&58&-24\\& & 16& \color{blue}{72} & & \\ \hline &2&\color{blue}{9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 72 } = \color{orangered}{ 55 } $
$$ \begin{array}{c|rrrrr}8&2&-7&\color{orangered}{ -17 }&58&-24\\& & 16& \color{orangered}{72} & & \\ \hline &2&9&\color{orangered}{55}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 55 } = \color{blue}{ 440 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&2&-7&-17&58&-24\\& & 16& 72& \color{blue}{440} & \\ \hline &2&9&\color{blue}{55}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 58 } + \color{orangered}{ 440 } = \color{orangered}{ 498 } $
$$ \begin{array}{c|rrrrr}8&2&-7&-17&\color{orangered}{ 58 }&-24\\& & 16& 72& \color{orangered}{440} & \\ \hline &2&9&55&\color{orangered}{498}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 498 } = \color{blue}{ 3984 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&2&-7&-17&58&-24\\& & 16& 72& 440& \color{blue}{3984} \\ \hline &2&9&55&\color{blue}{498}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 3984 } = \color{orangered}{ 3960 } $
$$ \begin{array}{c|rrrrr}8&2&-7&-17&58&\color{orangered}{ -24 }\\& & 16& 72& 440& \color{orangered}{3984} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{55}&\color{blue}{498}&\color{orangered}{3960} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+9x^{2}+55x+498 } $ with a remainder of $ \color{red}{ 3960 } $.