Find remainder when $ 4x^{3}+6x^{2}-x+15 $ is divided by $ x+3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&4&6&-1&15\\& & -12& 18& \color{black}{-51} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{17}&\color{orangered}{-36} \end{array} $$The remainder when $ 4x^{3}+6x^{2}-x+15 $ is divided by $ x+3 $ is $ \, \color{red}{ -36 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&6&-1&15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 4 }&6&-1&15\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&6&-1&15\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-3&4&\color{orangered}{ 6 }&-1&15\\& & \color{orangered}{-12} & & \\ \hline &4&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&6&-1&15\\& & -12& \color{blue}{18} & \\ \hline &4&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 18 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}-3&4&6&\color{orangered}{ -1 }&15\\& & -12& \color{orangered}{18} & \\ \hline &4&-6&\color{orangered}{17}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 17 } = \color{blue}{ -51 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&4&6&-1&15\\& & -12& 18& \color{blue}{-51} \\ \hline &4&-6&\color{blue}{17}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -51 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrr}-3&4&6&-1&\color{orangered}{ 15 }\\& & -12& 18& \color{orangered}{-51} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{17}&\color{orangered}{-36} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -36 }\right) $.