Find remainder when $ 2x^{3}-7x^{2}+8x-10 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&2&-7&8&-10\\& & 6& -3& \color{black}{15} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{5}&\color{orangered}{5} \end{array} $$The remainder when $ 2x^{3}-7x^{2}+8x-10 $ is divided by $ x-3 $ is $ \, \color{red}{ 5 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-7&8&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&-7&8&-10\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-7&8&-10\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 6 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ -7 }&8&-10\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-7&8&-10\\& & 6& \color{blue}{-3} & \\ \hline &2&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}3&2&-7&\color{orangered}{ 8 }&-10\\& & 6& \color{orangered}{-3} & \\ \hline &2&-1&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 5 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-7&8&-10\\& & 6& -3& \color{blue}{15} \\ \hline &2&-1&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 15 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}3&2&-7&8&\color{orangered}{ -10 }\\& & 6& -3& \color{orangered}{15} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{5}&\color{orangered}{5} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right) $.