Find remainder when $ 7x^{4}+6x^{3}-36x^{2}-30x+5 $ is divided by $ x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&7&6&-36&-30&5\\& & -7& 1& 35& \color{black}{-5} \\ \hline &\color{blue}{7}&\color{blue}{-1}&\color{blue}{-35}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The remainder when $ 7x^{4}+6x^{3}-36x^{2}-30x+5 $ is divided by $ x+1 $ is $ \, \color{red}{ 0 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&7&6&-36&-30&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 7 }&6&-36&-30&5\\& & & & & \\ \hline &\color{orangered}{7}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 7 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&7&6&-36&-30&5\\& & \color{blue}{-7} & & & \\ \hline &\color{blue}{7}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-1&7&\color{orangered}{ 6 }&-36&-30&5\\& & \color{orangered}{-7} & & & \\ \hline &7&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&7&6&-36&-30&5\\& & -7& \color{blue}{1} & & \\ \hline &7&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 1 } = \color{orangered}{ -35 } $
$$ \begin{array}{c|rrrrr}-1&7&6&\color{orangered}{ -36 }&-30&5\\& & -7& \color{orangered}{1} & & \\ \hline &7&-1&\color{orangered}{-35}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -35 \right) } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&7&6&-36&-30&5\\& & -7& 1& \color{blue}{35} & \\ \hline &7&-1&\color{blue}{-35}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 35 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-1&7&6&-36&\color{orangered}{ -30 }&5\\& & -7& 1& \color{orangered}{35} & \\ \hline &7&-1&-35&\color{orangered}{5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&7&6&-36&-30&5\\& & -7& 1& 35& \color{blue}{-5} \\ \hline &7&-1&-35&\color{blue}{5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-1&7&6&-36&-30&\color{orangered}{ 5 }\\& & -7& 1& 35& \color{orangered}{-5} \\ \hline &\color{blue}{7}&\color{blue}{-1}&\color{blue}{-35}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.