Divide using synthetic division $$ ( x^{4}+4x^{3}+24x+15 ) \div ( 3x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&1&4&0&24&15\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{0}&\color{blue}{24}&\color{orangered}{15} \end{array} $$The solution is:
$$ \frac{ x^{4}+4x^{3}+24x+15 }{ x } = \color{blue}{x^{3}+4x^{2}+24} ~+~ \frac{ \color{red}{ 15 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&4&0&24&15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&4&0&24&15\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&4&0&24&15\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ 4 }&0&24&15\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&4&0&24&15\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&1&4&\color{orangered}{ 0 }&24&15\\& & 0& \color{orangered}{0} & & \\ \hline &1&4&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&4&0&24&15\\& & 0& 0& \color{blue}{0} & \\ \hline &1&4&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 0 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}0&1&4&0&\color{orangered}{ 24 }&15\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&4&0&\color{orangered}{24}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 24 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&4&0&24&15\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&4&0&\color{blue}{24}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ 0 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}0&1&4&0&24&\color{orangered}{ 15 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{0}&\color{blue}{24}&\color{orangered}{15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+4x^{2}+24 } $ with a remainder of $ \color{red}{ 15 } $.