Find remainder when $ x^{3}-7x^{2}+15x-9 $ is divided by $ x+3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&-7&15&-9\\& & -3& 30& \color{black}{-135} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{45}&\color{orangered}{-144} \end{array} $$The remainder when $ x^{3}-7x^{2}+15x-9 $ is divided by $ x+3 $ is $ \, \color{red}{ -144 } $.
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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-7&15&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&-7&15&-9\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-7&15&-9\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ -7 }&15&-9\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-7&15&-9\\& & -3& \color{blue}{30} & \\ \hline &1&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ 30 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrr}-3&1&-7&\color{orangered}{ 15 }&-9\\& & -3& \color{orangered}{30} & \\ \hline &1&-10&\color{orangered}{45}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 45 } = \color{blue}{ -135 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-7&15&-9\\& & -3& 30& \color{blue}{-135} \\ \hline &1&-10&\color{blue}{45}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -135 \right) } = \color{orangered}{ -144 } $
$$ \begin{array}{c|rrrr}-3&1&-7&15&\color{orangered}{ -9 }\\& & -3& 30& \color{orangered}{-135} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{45}&\color{orangered}{-144} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -144 }\right) $.