Divide using synthetic division $$ ( 16x^{3}+24x^{2}+29x+6 ) \div ( 4x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&16&24&29&6\\& & -16& -8& \color{black}{-21} \\ \hline &\color{blue}{16}&\color{blue}{8}&\color{blue}{21}&\color{orangered}{-15} \end{array} $$The solution is:
$$ \frac{ 16x^{3}+24x^{2}+29x+6 }{ x+1 } = \color{blue}{16x^{2}+8x+21} \color{red}{~-~} \frac{ \color{red}{ 15 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&24&29&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 16 }&24&29&6\\& & & & \\ \hline &\color{orangered}{16}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 16 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&24&29&6\\& & \color{blue}{-16} & & \\ \hline &\color{blue}{16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}-1&16&\color{orangered}{ 24 }&29&6\\& & \color{orangered}{-16} & & \\ \hline &16&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 8 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&24&29&6\\& & -16& \color{blue}{-8} & \\ \hline &16&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}-1&16&24&\color{orangered}{ 29 }&6\\& & -16& \color{orangered}{-8} & \\ \hline &16&8&\color{orangered}{21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 21 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&24&29&6\\& & -16& -8& \color{blue}{-21} \\ \hline &16&8&\color{blue}{21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}-1&16&24&29&\color{orangered}{ 6 }\\& & -16& -8& \color{orangered}{-21} \\ \hline &\color{blue}{16}&\color{blue}{8}&\color{blue}{21}&\color{orangered}{-15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 16x^{2}+8x+21 } $ with a remainder of $ \color{red}{ -15 } $.