Find remainder when $ 3x^{3}-7x^{2}+6x-14 $ is divided by $ x-7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&3&-7&6&-14\\& & 21& 98& \color{black}{728} \\ \hline &\color{blue}{3}&\color{blue}{14}&\color{blue}{104}&\color{orangered}{714} \end{array} $$The remainder when $ 3x^{3}-7x^{2}+6x-14 $ is divided by $ x-7 $ is $ \, \color{red}{ 714 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&-7&6&-14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 3 }&-7&6&-14\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 3 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&-7&6&-14\\& & \color{blue}{21} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 21 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}7&3&\color{orangered}{ -7 }&6&-14\\& & \color{orangered}{21} & & \\ \hline &3&\color{orangered}{14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 14 } = \color{blue}{ 98 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&-7&6&-14\\& & 21& \color{blue}{98} & \\ \hline &3&\color{blue}{14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 98 } = \color{orangered}{ 104 } $
$$ \begin{array}{c|rrrr}7&3&-7&\color{orangered}{ 6 }&-14\\& & 21& \color{orangered}{98} & \\ \hline &3&14&\color{orangered}{104}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 104 } = \color{blue}{ 728 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&-7&6&-14\\& & 21& 98& \color{blue}{728} \\ \hline &3&14&\color{blue}{104}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 728 } = \color{orangered}{ 714 } $
$$ \begin{array}{c|rrrr}7&3&-7&6&\color{orangered}{ -14 }\\& & 21& 98& \color{orangered}{728} \\ \hline &\color{blue}{3}&\color{blue}{14}&\color{blue}{104}&\color{orangered}{714} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 714 }\right) $.