Find remainder when $ 13x^{6}+33x^{5}+9x^{4}-13x^{3}+11x-2 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&13&33&9&-13&0&11&-2\\& & -26& -14& 10& 6& -12& \color{black}{2} \\ \hline &\color{blue}{13}&\color{blue}{7}&\color{blue}{-5}&\color{blue}{-3}&\color{blue}{6}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$The remainder when $ 13x^{6}+33x^{5}+9x^{4}-13x^{3}+11x-2 $ is divided by $ x+2 $ 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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 13 }&33&9&-13&0&11&-2\\& & & & & & & \\ \hline &\color{orangered}{13}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 13 } = \color{blue}{ -26 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & \color{blue}{-26} & & & & & \\ \hline &\color{blue}{13}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ \left( -26 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrrr}-2&13&\color{orangered}{ 33 }&9&-13&0&11&-2\\& & \color{orangered}{-26} & & & & & \\ \hline &13&\color{orangered}{7}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & -26& \color{blue}{-14} & & & & \\ \hline &13&\color{blue}{7}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrr}-2&13&33&\color{orangered}{ 9 }&-13&0&11&-2\\& & -26& \color{orangered}{-14} & & & & \\ \hline &13&7&\color{orangered}{-5}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & -26& -14& \color{blue}{10} & & & \\ \hline &13&7&\color{blue}{-5}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 10 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrr}-2&13&33&9&\color{orangered}{ -13 }&0&11&-2\\& & -26& -14& \color{orangered}{10} & & & \\ \hline &13&7&-5&\color{orangered}{-3}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & -26& -14& 10& \color{blue}{6} & & \\ \hline &13&7&-5&\color{blue}{-3}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrr}-2&13&33&9&-13&\color{orangered}{ 0 }&11&-2\\& & -26& -14& 10& \color{orangered}{6} & & \\ \hline &13&7&-5&-3&\color{orangered}{6}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & -26& -14& 10& 6& \color{blue}{-12} & \\ \hline &13&7&-5&-3&\color{blue}{6}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}-2&13&33&9&-13&0&\color{orangered}{ 11 }&-2\\& & -26& -14& 10& 6& \color{orangered}{-12} & \\ \hline &13&7&-5&-3&6&\color{orangered}{-1}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&13&33&9&-13&0&11&-2\\& & -26& -14& 10& 6& -12& \color{blue}{2} \\ \hline &13&7&-5&-3&6&\color{blue}{-1}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 2 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-2&13&33&9&-13&0&11&\color{orangered}{ -2 }\\& & -26& -14& 10& 6& -12& \color{orangered}{2} \\ \hline &\color{blue}{13}&\color{blue}{7}&\color{blue}{-5}&\color{blue}{-3}&\color{blue}{6}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.