Divide using synthetic division $$ ( 4x^{5}-x^{4}+2x^{3}+3x-3 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&4&-1&2&0&3&-3\\& & -16& 68& -280& 1120& \color{black}{-4492} \\ \hline &\color{blue}{4}&\color{blue}{-17}&\color{blue}{70}&\color{blue}{-280}&\color{blue}{1123}&\color{orangered}{-4495} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-x^{4}+2x^{3}+3x-3 }{ x+4 } = \color{blue}{4x^{4}-17x^{3}+70x^{2}-280x+1123} \color{red}{~-~} \frac{ \color{red}{ 4495 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 4 }&-1&2&0&3&-3\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & \color{blue}{-16} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrr}-4&4&\color{orangered}{ -1 }&2&0&3&-3\\& & \color{orangered}{-16} & & & & \\ \hline &4&\color{orangered}{-17}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 68 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & -16& \color{blue}{68} & & & \\ \hline &4&\color{blue}{-17}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 68 } = \color{orangered}{ 70 } $
$$ \begin{array}{c|rrrrrr}-4&4&-1&\color{orangered}{ 2 }&0&3&-3\\& & -16& \color{orangered}{68} & & & \\ \hline &4&-17&\color{orangered}{70}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 70 } = \color{blue}{ -280 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & -16& 68& \color{blue}{-280} & & \\ \hline &4&-17&\color{blue}{70}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -280 \right) } = \color{orangered}{ -280 } $
$$ \begin{array}{c|rrrrrr}-4&4&-1&2&\color{orangered}{ 0 }&3&-3\\& & -16& 68& \color{orangered}{-280} & & \\ \hline &4&-17&70&\color{orangered}{-280}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -280 \right) } = \color{blue}{ 1120 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & -16& 68& -280& \color{blue}{1120} & \\ \hline &4&-17&70&\color{blue}{-280}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 1120 } = \color{orangered}{ 1123 } $
$$ \begin{array}{c|rrrrrr}-4&4&-1&2&0&\color{orangered}{ 3 }&-3\\& & -16& 68& -280& \color{orangered}{1120} & \\ \hline &4&-17&70&-280&\color{orangered}{1123}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1123 } = \color{blue}{ -4492 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&-1&2&0&3&-3\\& & -16& 68& -280& 1120& \color{blue}{-4492} \\ \hline &4&-17&70&-280&\color{blue}{1123}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -4492 \right) } = \color{orangered}{ -4495 } $
$$ \begin{array}{c|rrrrrr}-4&4&-1&2&0&3&\color{orangered}{ -3 }\\& & -16& 68& -280& 1120& \color{orangered}{-4492} \\ \hline &\color{blue}{4}&\color{blue}{-17}&\color{blue}{70}&\color{blue}{-280}&\color{blue}{1123}&\color{orangered}{-4495} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}-17x^{3}+70x^{2}-280x+1123 } $ with a remainder of $ \color{red}{ -4495 } $.