Divide using synthetic division $$ ( x^{4}-5x^{3}-10x+51 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&1&-5&0&-10&51\\& & 5& 0& 0& \color{black}{-50} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-10}&\color{orangered}{1} \end{array} $$The solution is:
$$ \frac{ x^{4}-5x^{3}-10x+51 }{ x-5 } = \color{blue}{x^{3}-10} ~+~ \frac{ \color{red}{ 1 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&1&-5&0&-10&51\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 1 }&-5&0&-10&51\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&1&-5&0&-10&51\\& & \color{blue}{5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 5 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&1&\color{orangered}{ -5 }&0&-10&51\\& & \color{orangered}{5} & & & \\ \hline &1&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&1&-5&0&-10&51\\& & 5& \color{blue}{0} & & \\ \hline &1&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&1&-5&\color{orangered}{ 0 }&-10&51\\& & 5& \color{orangered}{0} & & \\ \hline &1&0&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&1&-5&0&-10&51\\& & 5& 0& \color{blue}{0} & \\ \hline &1&0&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}5&1&-5&0&\color{orangered}{ -10 }&51\\& & 5& 0& \color{orangered}{0} & \\ \hline &1&0&0&\color{orangered}{-10}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&1&-5&0&-10&51\\& & 5& 0& 0& \color{blue}{-50} \\ \hline &1&0&0&\color{blue}{-10}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 51 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}5&1&-5&0&-10&\color{orangered}{ 51 }\\& & 5& 0& 0& \color{orangered}{-50} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-10}&\color{orangered}{1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-10 } $ with a remainder of $ \color{red}{ 1 } $.