Divide using synthetic division $$ ( 3x^{4}-2x^{2}+5 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&3&0&-2&0&5\\& & -9& 27& -75& \color{black}{225} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{25}&\color{blue}{-75}&\color{orangered}{230} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-2x^{2}+5 }{ x+3 } = \color{blue}{3x^{3}-9x^{2}+25x-75} ~+~ \frac{ \color{red}{ 230 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&0&-2&0&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 3 }&0&-2&0&5\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&0&-2&0&5\\& & \color{blue}{-9} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-3&3&\color{orangered}{ 0 }&-2&0&5\\& & \color{orangered}{-9} & & & \\ \hline &3&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&0&-2&0&5\\& & -9& \color{blue}{27} & & \\ \hline &3&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 27 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrr}-3&3&0&\color{orangered}{ -2 }&0&5\\& & -9& \color{orangered}{27} & & \\ \hline &3&-9&\color{orangered}{25}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 25 } = \color{blue}{ -75 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&0&-2&0&5\\& & -9& 27& \color{blue}{-75} & \\ \hline &3&-9&\color{blue}{25}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -75 \right) } = \color{orangered}{ -75 } $
$$ \begin{array}{c|rrrrr}-3&3&0&-2&\color{orangered}{ 0 }&5\\& & -9& 27& \color{orangered}{-75} & \\ \hline &3&-9&25&\color{orangered}{-75}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -75 \right) } = \color{blue}{ 225 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&0&-2&0&5\\& & -9& 27& -75& \color{blue}{225} \\ \hline &3&-9&25&\color{blue}{-75}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 225 } = \color{orangered}{ 230 } $
$$ \begin{array}{c|rrrrr}-3&3&0&-2&0&\color{orangered}{ 5 }\\& & -9& 27& -75& \color{orangered}{225} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{25}&\color{blue}{-75}&\color{orangered}{230} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-9x^{2}+25x-75 } $ with a remainder of $ \color{red}{ 230 } $.