Divide using synthetic division $$ ( 7x^{5}-3x+3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&7&0&0&0&-3&3\\& & -14& 28& -56& 112& \color{black}{-218} \\ \hline &\color{blue}{7}&\color{blue}{-14}&\color{blue}{28}&\color{blue}{-56}&\color{blue}{109}&\color{orangered}{-215} \end{array} $$The solution is:
$$ \frac{ 7x^{5}-3x+3 }{ x+2 } = \color{blue}{7x^{4}-14x^{3}+28x^{2}-56x+109} \color{red}{~-~} \frac{ \color{red}{ 215 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 7 }&0&0&0&-3&3\\& & & & & & \\ \hline &\color{orangered}{7}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & \color{blue}{-14} & & & & \\ \hline &\color{blue}{7}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrrrr}-2&7&\color{orangered}{ 0 }&0&0&-3&3\\& & \color{orangered}{-14} & & & & \\ \hline &7&\color{orangered}{-14}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & -14& \color{blue}{28} & & & \\ \hline &7&\color{blue}{-14}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 28 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrrr}-2&7&0&\color{orangered}{ 0 }&0&-3&3\\& & -14& \color{orangered}{28} & & & \\ \hline &7&-14&\color{orangered}{28}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 28 } = \color{blue}{ -56 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & -14& 28& \color{blue}{-56} & & \\ \hline &7&-14&\color{blue}{28}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -56 \right) } = \color{orangered}{ -56 } $
$$ \begin{array}{c|rrrrrr}-2&7&0&0&\color{orangered}{ 0 }&-3&3\\& & -14& 28& \color{orangered}{-56} & & \\ \hline &7&-14&28&\color{orangered}{-56}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -56 \right) } = \color{blue}{ 112 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & -14& 28& -56& \color{blue}{112} & \\ \hline &7&-14&28&\color{blue}{-56}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 112 } = \color{orangered}{ 109 } $
$$ \begin{array}{c|rrrrrr}-2&7&0&0&0&\color{orangered}{ -3 }&3\\& & -14& 28& -56& \color{orangered}{112} & \\ \hline &7&-14&28&-56&\color{orangered}{109}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 109 } = \color{blue}{ -218 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&7&0&0&0&-3&3\\& & -14& 28& -56& 112& \color{blue}{-218} \\ \hline &7&-14&28&-56&\color{blue}{109}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -218 \right) } = \color{orangered}{ -215 } $
$$ \begin{array}{c|rrrrrr}-2&7&0&0&0&-3&\color{orangered}{ 3 }\\& & -14& 28& -56& 112& \color{orangered}{-218} \\ \hline &\color{blue}{7}&\color{blue}{-14}&\color{blue}{28}&\color{blue}{-56}&\color{blue}{109}&\color{orangered}{-215} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x^{4}-14x^{3}+28x^{2}-56x+109 } $ with a remainder of $ \color{red}{ -215 } $.