Divide using synthetic division $$ ( 4x^{5}-3x^{3}+2x^{2}-2x+1 ) \div ( 2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&4&0&-3&2&-2&1\\& & -12& 36& -99& 291& \color{black}{-867} \\ \hline &\color{blue}{4}&\color{blue}{-12}&\color{blue}{33}&\color{blue}{-97}&\color{blue}{289}&\color{orangered}{-866} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-3x^{3}+2x^{2}-2x+1 }{ x+3 } = \color{blue}{4x^{4}-12x^{3}+33x^{2}-97x+289} \color{red}{~-~} \frac{ \color{red}{ 866 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 4 }&0&-3&2&-2&1\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 4 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & \color{blue}{-12} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrr}-3&4&\color{orangered}{ 0 }&-3&2&-2&1\\& & \color{orangered}{-12} & & & & \\ \hline &4&\color{orangered}{-12}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & -12& \color{blue}{36} & & & \\ \hline &4&\color{blue}{-12}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 36 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrrrr}-3&4&0&\color{orangered}{ -3 }&2&-2&1\\& & -12& \color{orangered}{36} & & & \\ \hline &4&-12&\color{orangered}{33}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 33 } = \color{blue}{ -99 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & -12& 36& \color{blue}{-99} & & \\ \hline &4&-12&\color{blue}{33}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -99 \right) } = \color{orangered}{ -97 } $
$$ \begin{array}{c|rrrrrr}-3&4&0&-3&\color{orangered}{ 2 }&-2&1\\& & -12& 36& \color{orangered}{-99} & & \\ \hline &4&-12&33&\color{orangered}{-97}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -97 \right) } = \color{blue}{ 291 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & -12& 36& -99& \color{blue}{291} & \\ \hline &4&-12&33&\color{blue}{-97}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 291 } = \color{orangered}{ 289 } $
$$ \begin{array}{c|rrrrrr}-3&4&0&-3&2&\color{orangered}{ -2 }&1\\& & -12& 36& -99& \color{orangered}{291} & \\ \hline &4&-12&33&-97&\color{orangered}{289}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 289 } = \color{blue}{ -867 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&0&-3&2&-2&1\\& & -12& 36& -99& 291& \color{blue}{-867} \\ \hline &4&-12&33&-97&\color{blue}{289}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -867 \right) } = \color{orangered}{ -866 } $
$$ \begin{array}{c|rrrrrr}-3&4&0&-3&2&-2&\color{orangered}{ 1 }\\& & -12& 36& -99& 291& \color{orangered}{-867} \\ \hline &\color{blue}{4}&\color{blue}{-12}&\color{blue}{33}&\color{blue}{-97}&\color{blue}{289}&\color{orangered}{-866} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}-12x^{3}+33x^{2}-97x+289 } $ with a remainder of $ \color{red}{ -866 } $.