Divide using synthetic division $$ ( 2x^{5}-3x^{4}-9x^{3}+8x^{2}+2 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&2&-3&-9&8&0&2\\& & 6& 9& 0& 24& \color{black}{72} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{0}&\color{blue}{8}&\color{blue}{24}&\color{orangered}{74} \end{array} $$The solution is:
$$ \frac{ 2x^{5}-3x^{4}-9x^{3}+8x^{2}+2 }{ x-3 } = \color{blue}{2x^{4}+3x^{3}+8x+24} ~+~ \frac{ \color{red}{ 74 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 2 }&-3&-9&8&0&2\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 6 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}3&2&\color{orangered}{ -3 }&-9&8&0&2\\& & \color{orangered}{6} & & & & \\ \hline &2&\color{orangered}{3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & 6& \color{blue}{9} & & & \\ \hline &2&\color{blue}{3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 9 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}3&2&-3&\color{orangered}{ -9 }&8&0&2\\& & 6& \color{orangered}{9} & & & \\ \hline &2&3&\color{orangered}{0}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & 6& 9& \color{blue}{0} & & \\ \hline &2&3&\color{blue}{0}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}3&2&-3&-9&\color{orangered}{ 8 }&0&2\\& & 6& 9& \color{orangered}{0} & & \\ \hline &2&3&0&\color{orangered}{8}&& \end{array} $$Step 8 : 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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & 6& 9& 0& \color{blue}{24} & \\ \hline &2&3&0&\color{blue}{8}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrrr}3&2&-3&-9&8&\color{orangered}{ 0 }&2\\& & 6& 9& 0& \color{orangered}{24} & \\ \hline &2&3&0&8&\color{orangered}{24}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 24 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&-3&-9&8&0&2\\& & 6& 9& 0& 24& \color{blue}{72} \\ \hline &2&3&0&8&\color{blue}{24}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 72 } = \color{orangered}{ 74 } $
$$ \begin{array}{c|rrrrrr}3&2&-3&-9&8&0&\color{orangered}{ 2 }\\& & 6& 9& 0& 24& \color{orangered}{72} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{0}&\color{blue}{8}&\color{blue}{24}&\color{orangered}{74} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}+3x^{3}+8x+24 } $ with a remainder of $ \color{red}{ 74 } $.