Divide using synthetic division $$ ( 3x^{7}+5x^{4}-x+3 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&5&0&0&-1&3\\& & -3& 3& -3& -2& 2& -2& \color{black}{3} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{3}&\color{blue}{2}&\color{blue}{-2}&\color{blue}{2}&\color{blue}{-3}&\color{orangered}{6} \end{array} $$The solution is:
$$ \frac{ 3x^{7}+5x^{4}-x+3 }{ x+1 } = \color{blue}{3x^{6}-3x^{5}+3x^{4}+2x^{3}-2x^{2}+2x-3} ~+~ \frac{ \color{red}{ 6 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-1&\color{orangered}{ 3 }&0&0&5&0&0&-1&3\\& & & & & & & & \\ \hline &\color{orangered}{3}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & \color{blue}{-3} & & & & & & \\ \hline &\color{blue}{3}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&\color{orangered}{ 0 }&0&5&0&0&-1&3\\& & \color{orangered}{-3} & & & & & & \\ \hline &3&\color{orangered}{-3}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& \color{blue}{3} & & & & & \\ \hline &3&\color{blue}{-3}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 3 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&\color{orangered}{ 0 }&5&0&0&-1&3\\& & -3& \color{orangered}{3} & & & & & \\ \hline &3&-3&\color{orangered}{3}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& 3& \color{blue}{-3} & & & & \\ \hline &3&-3&\color{blue}{3}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&\color{orangered}{ 5 }&0&0&-1&3\\& & -3& 3& \color{orangered}{-3} & & & & \\ \hline &3&-3&3&\color{orangered}{2}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& 3& -3& \color{blue}{-2} & & & \\ \hline &3&-3&3&\color{blue}{2}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&5&\color{orangered}{ 0 }&0&-1&3\\& & -3& 3& -3& \color{orangered}{-2} & & & \\ \hline &3&-3&3&2&\color{orangered}{-2}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& 3& -3& -2& \color{blue}{2} & & \\ \hline &3&-3&3&2&\color{blue}{-2}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&5&0&\color{orangered}{ 0 }&-1&3\\& & -3& 3& -3& -2& \color{orangered}{2} & & \\ \hline &3&-3&3&2&-2&\color{orangered}{2}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& 3& -3& -2& 2& \color{blue}{-2} & \\ \hline &3&-3&3&2&-2&\color{blue}{2}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&5&0&0&\color{orangered}{ -1 }&3\\& & -3& 3& -3& -2& 2& \color{orangered}{-2} & \\ \hline &3&-3&3&2&-2&2&\color{orangered}{-3}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&3&0&0&5&0&0&-1&3\\& & -3& 3& -3& -2& 2& -2& \color{blue}{3} \\ \hline &3&-3&3&2&-2&2&\color{blue}{-3}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrr}-1&3&0&0&5&0&0&-1&\color{orangered}{ 3 }\\& & -3& 3& -3& -2& 2& -2& \color{orangered}{3} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{3}&\color{blue}{2}&\color{blue}{-2}&\color{blue}{2}&\color{blue}{-3}&\color{orangered}{6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{6}-3x^{5}+3x^{4}+2x^{3}-2x^{2}+2x-3 } $ with a remainder of $ \color{red}{ 6 } $.