Divide using synthetic division $$ ( -5x^{5}+30x^{4}+3x^{3}-17x^{2}-10x+24 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}6&-5&30&3&-17&-10&24\\& & -30& 0& 18& 6& \color{black}{-24} \\ \hline &\color{blue}{-5}&\color{blue}{0}&\color{blue}{3}&\color{blue}{1}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -5x^{5}+30x^{4}+3x^{3}-17x^{2}-10x+24 }{ x-6 } = \color{blue}{-5x^{4}+3x^{2}+x-4} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}6&\color{orangered}{ -5 }&30&3&-17&-10&24\\& & & & & & \\ \hline &\color{orangered}{-5}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & \color{blue}{-30} & & & & \\ \hline &\color{blue}{-5}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}6&-5&\color{orangered}{ 30 }&3&-17&-10&24\\& & \color{orangered}{-30} & & & & \\ \hline &-5&\color{orangered}{0}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & -30& \color{blue}{0} & & & \\ \hline &-5&\color{blue}{0}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}6&-5&30&\color{orangered}{ 3 }&-17&-10&24\\& & -30& \color{orangered}{0} & & & \\ \hline &-5&0&\color{orangered}{3}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 3 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & -30& 0& \color{blue}{18} & & \\ \hline &-5&0&\color{blue}{3}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 18 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}6&-5&30&3&\color{orangered}{ -17 }&-10&24\\& & -30& 0& \color{orangered}{18} & & \\ \hline &-5&0&3&\color{orangered}{1}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & -30& 0& 18& \color{blue}{6} & \\ \hline &-5&0&3&\color{blue}{1}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 6 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}6&-5&30&3&-17&\color{orangered}{ -10 }&24\\& & -30& 0& 18& \color{orangered}{6} & \\ \hline &-5&0&3&1&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&-5&30&3&-17&-10&24\\& & -30& 0& 18& 6& \color{blue}{-24} \\ \hline &-5&0&3&1&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}6&-5&30&3&-17&-10&\color{orangered}{ 24 }\\& & -30& 0& 18& 6& \color{orangered}{-24} \\ \hline &\color{blue}{-5}&\color{blue}{0}&\color{blue}{3}&\color{blue}{1}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{4}+3x^{2}+x-4 } $ with a remainder of $ \color{red}{ 0 } $.