Find remainder when $ x^{3}-15x^{2}+68x-84 $ is divided by $ x-7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&1&-15&68&-84\\& & 7& -56& \color{black}{84} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{12}&\color{orangered}{0} \end{array} $$The remainder when $ x^{3}-15x^{2}+68x-84 $ is divided by $ x-7 $ is $ \, \color{red}{ 0 } $.
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 -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-15&68&-84\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 1 }&-15&68&-84\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-15&68&-84\\& & \color{blue}{7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 7 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}7&1&\color{orangered}{ -15 }&68&-84\\& & \color{orangered}{7} & & \\ \hline &1&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -56 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-15&68&-84\\& & 7& \color{blue}{-56} & \\ \hline &1&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 68 } + \color{orangered}{ \left( -56 \right) } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}7&1&-15&\color{orangered}{ 68 }&-84\\& & 7& \color{orangered}{-56} & \\ \hline &1&-8&\color{orangered}{12}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 12 } = \color{blue}{ 84 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-15&68&-84\\& & 7& -56& \color{blue}{84} \\ \hline &1&-8&\color{blue}{12}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -84 } + \color{orangered}{ 84 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}7&1&-15&68&\color{orangered}{ -84 }\\& & 7& -56& \color{orangered}{84} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{12}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.