Divide using synthetic division $$ ( x^{4}-6x^{3}+11x^{2}+29x-13 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&-6&11&29&-13\\& & -5& 55& -330& \color{black}{1505} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{66}&\color{blue}{-301}&\color{orangered}{1492} \end{array} $$The solution is:
$$ \frac{ x^{4}-6x^{3}+11x^{2}+29x-13 }{ x+5 } = \color{blue}{x^{3}-11x^{2}+66x-301} ~+~ \frac{ \color{red}{ 1492 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&-6&11&29&-13\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&-6&11&29&-13\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&-6&11&29&-13\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ -6 }&11&29&-13\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{-11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 55 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&-6&11&29&-13\\& & -5& \color{blue}{55} & & \\ \hline &1&\color{blue}{-11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 55 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrrr}-5&1&-6&\color{orangered}{ 11 }&29&-13\\& & -5& \color{orangered}{55} & & \\ \hline &1&-11&\color{orangered}{66}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 66 } = \color{blue}{ -330 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&-6&11&29&-13\\& & -5& 55& \color{blue}{-330} & \\ \hline &1&-11&\color{blue}{66}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -330 \right) } = \color{orangered}{ -301 } $
$$ \begin{array}{c|rrrrr}-5&1&-6&11&\color{orangered}{ 29 }&-13\\& & -5& 55& \color{orangered}{-330} & \\ \hline &1&-11&66&\color{orangered}{-301}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -301 \right) } = \color{blue}{ 1505 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&-6&11&29&-13\\& & -5& 55& -330& \color{blue}{1505} \\ \hline &1&-11&66&\color{blue}{-301}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 1505 } = \color{orangered}{ 1492 } $
$$ \begin{array}{c|rrrrr}-5&1&-6&11&29&\color{orangered}{ -13 }\\& & -5& 55& -330& \color{orangered}{1505} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{66}&\color{blue}{-301}&\color{orangered}{1492} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-11x^{2}+66x-301 } $ with a remainder of $ \color{red}{ 1492 } $.