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