Find remainder when $ 6x^{3}+13x^{2}-7x-15 $ is divided by $ 3x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&6&13&-7&-15\\& & -30& 85& \color{black}{-390} \\ \hline &\color{blue}{6}&\color{blue}{-17}&\color{blue}{78}&\color{orangered}{-405} \end{array} $$The remainder when $ 6x^{3}+13x^{2}-7x-15 $ is divided by $ x+5 $ is $ \, \color{red}{ -405 } $.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&13&-7&-15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 6 }&13&-7&-15\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 6 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&13&-7&-15\\& & \color{blue}{-30} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}-5&6&\color{orangered}{ 13 }&-7&-15\\& & \color{orangered}{-30} & & \\ \hline &6&\color{orangered}{-17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 85 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&13&-7&-15\\& & -30& \color{blue}{85} & \\ \hline &6&\color{blue}{-17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 85 } = \color{orangered}{ 78 } $
$$ \begin{array}{c|rrrr}-5&6&13&\color{orangered}{ -7 }&-15\\& & -30& \color{orangered}{85} & \\ \hline &6&-17&\color{orangered}{78}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 78 } = \color{blue}{ -390 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&13&-7&-15\\& & -30& 85& \color{blue}{-390} \\ \hline &6&-17&\color{blue}{78}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -390 \right) } = \color{orangered}{ -405 } $
$$ \begin{array}{c|rrrr}-5&6&13&-7&\color{orangered}{ -15 }\\& & -30& 85& \color{orangered}{-390} \\ \hline &\color{blue}{6}&\color{blue}{-17}&\color{blue}{78}&\color{orangered}{-405} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -405 }\right) $.