Find remainder when $ x^{4}-9x^{3}+21x^{2}+21x-130 $ is divided by $ -2x+3 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-9&21&21&-130\\& & -3& 36& -171& \color{black}{450} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{57}&\color{blue}{-150}&\color{orangered}{320} \end{array} $$The remainder when $ x^{4}-9x^{3}+21x^{2}+21x-130 $ is divided by $ x+3 $ is $ \, \color{red}{ 320 } $.
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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-9&21&21&-130\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-9&21&21&-130\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-9&21&21&-130\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -9 }&21&21&-130\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-12}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-9&21&21&-130\\& & -3& \color{blue}{36} & & \\ \hline &1&\color{blue}{-12}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ 36 } = \color{orangered}{ 57 } $
$$ \begin{array}{c|rrrrr}-3&1&-9&\color{orangered}{ 21 }&21&-130\\& & -3& \color{orangered}{36} & & \\ \hline &1&-12&\color{orangered}{57}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 57 } = \color{blue}{ -171 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-9&21&21&-130\\& & -3& 36& \color{blue}{-171} & \\ \hline &1&-12&\color{blue}{57}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -171 \right) } = \color{orangered}{ -150 } $
$$ \begin{array}{c|rrrrr}-3&1&-9&21&\color{orangered}{ 21 }&-130\\& & -3& 36& \color{orangered}{-171} & \\ \hline &1&-12&57&\color{orangered}{-150}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -150 \right) } = \color{blue}{ 450 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-9&21&21&-130\\& & -3& 36& -171& \color{blue}{450} \\ \hline &1&-12&57&\color{blue}{-150}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -130 } + \color{orangered}{ 450 } = \color{orangered}{ 320 } $
$$ \begin{array}{c|rrrrr}-3&1&-9&21&21&\color{orangered}{ -130 }\\& & -3& 36& -171& \color{orangered}{450} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{57}&\color{blue}{-150}&\color{orangered}{320} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 320 }\right) $.