Divide using synthetic division $$ ( 24x^{5}+30x^{4}-25x^{2}+21 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-1&24&30&0&-25&0&21\\& & -24& -6& 6& 19& \color{black}{-19} \\ \hline &\color{blue}{24}&\color{blue}{6}&\color{blue}{-6}&\color{blue}{-19}&\color{blue}{19}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ 24x^{5}+30x^{4}-25x^{2}+21 }{ x+1 } = \color{blue}{24x^{4}+6x^{3}-6x^{2}-19x+19} ~+~ \frac{ \color{red}{ 2 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ 24 }&30&0&-25&0&21\\& & & & & & \\ \hline &\color{orangered}{24}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 24 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & \color{blue}{-24} & & & & \\ \hline &\color{blue}{24}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrr}-1&24&\color{orangered}{ 30 }&0&-25&0&21\\& & \color{orangered}{-24} & & & & \\ \hline &24&\color{orangered}{6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 6 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & -24& \color{blue}{-6} & & & \\ \hline &24&\color{blue}{6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-1&24&30&\color{orangered}{ 0 }&-25&0&21\\& & -24& \color{orangered}{-6} & & & \\ \hline &24&6&\color{orangered}{-6}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & -24& -6& \color{blue}{6} & & \\ \hline &24&6&\color{blue}{-6}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 6 } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrr}-1&24&30&0&\color{orangered}{ -25 }&0&21\\& & -24& -6& \color{orangered}{6} & & \\ \hline &24&6&-6&\color{orangered}{-19}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & -24& -6& 6& \color{blue}{19} & \\ \hline &24&6&-6&\color{blue}{-19}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 19 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrr}-1&24&30&0&-25&\color{orangered}{ 0 }&21\\& & -24& -6& 6& \color{orangered}{19} & \\ \hline &24&6&-6&-19&\color{orangered}{19}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 19 } = \color{blue}{ -19 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&24&30&0&-25&0&21\\& & -24& -6& 6& 19& \color{blue}{-19} \\ \hline &24&6&-6&-19&\color{blue}{19}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -19 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-1&24&30&0&-25&0&\color{orangered}{ 21 }\\& & -24& -6& 6& 19& \color{orangered}{-19} \\ \hline &\color{blue}{24}&\color{blue}{6}&\color{blue}{-6}&\color{blue}{-19}&\color{blue}{19}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 24x^{4}+6x^{3}-6x^{2}-19x+19 } $ with a remainder of $ \color{red}{ 2 } $.