Find remainder when $ 4x^{3}-15x^{2}+24x-17 $ is divided by $ x+3 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&4&-15&24&-17\\& & -12& 81& \color{black}{-315} \\ \hline &\color{blue}{4}&\color{blue}{-27}&\color{blue}{105}&\color{orangered}{-332} \end{array} $$The remainder when $ 4x^{3}-15x^{2}+24x-17 $ is divided by $ x+3 $ is $ \, \color{red}{ -332 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&-15&24&-17\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 4 }&-15&24&-17\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 4 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&-15&24&-17\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrr}-3&4&\color{orangered}{ -15 }&24&-17\\& & \color{orangered}{-12} & & \\ \hline &4&\color{orangered}{-27}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&-15&24&-17\\& & -12& \color{blue}{81} & \\ \hline &4&\color{blue}{-27}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 81 } = \color{orangered}{ 105 } $
$$ \begin{array}{c|rrrr}-3&4&-15&\color{orangered}{ 24 }&-17\\& & -12& \color{orangered}{81} & \\ \hline &4&-27&\color{orangered}{105}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 105 } = \color{blue}{ -315 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&-15&24&-17\\& & -12& 81& \color{blue}{-315} \\ \hline &4&-27&\color{blue}{105}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -315 \right) } = \color{orangered}{ -332 } $
$$ \begin{array}{c|rrrr}-3&4&-15&24&\color{orangered}{ -17 }\\& & -12& 81& \color{orangered}{-315} \\ \hline &\color{blue}{4}&\color{blue}{-27}&\color{blue}{105}&\color{orangered}{-332} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -332 }\right) $.