Find remainder when $ 11x^{4}-40x^{3}+24x^{2}-16x+21 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&11&-40&24&-16&21\\& & 33& -21& 9& \color{black}{-21} \\ \hline &\color{blue}{11}&\color{blue}{-7}&\color{blue}{3}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The remainder when $ 11x^{4}-40x^{3}+24x^{2}-16x+21 $ is divided by $ x-3 $ 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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&-40&24&-16&21\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 11 }&-40&24&-16&21\\& & & & & \\ \hline &\color{orangered}{11}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 11 } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&-40&24&-16&21\\& & \color{blue}{33} & & & \\ \hline &\color{blue}{11}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ 33 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}3&11&\color{orangered}{ -40 }&24&-16&21\\& & \color{orangered}{33} & & & \\ \hline &11&\color{orangered}{-7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&-40&24&-16&21\\& & 33& \color{blue}{-21} & & \\ \hline &11&\color{blue}{-7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}3&11&-40&\color{orangered}{ 24 }&-16&21\\& & 33& \color{orangered}{-21} & & \\ \hline &11&-7&\color{orangered}{3}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&-40&24&-16&21\\& & 33& -21& \color{blue}{9} & \\ \hline &11&-7&\color{blue}{3}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 9 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}3&11&-40&24&\color{orangered}{ -16 }&21\\& & 33& -21& \color{orangered}{9} & \\ \hline &11&-7&3&\color{orangered}{-7}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&-40&24&-16&21\\& & 33& -21& 9& \color{blue}{-21} \\ \hline &11&-7&3&\color{blue}{-7}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}3&11&-40&24&-16&\color{orangered}{ 21 }\\& & 33& -21& 9& \color{orangered}{-21} \\ \hline &\color{blue}{11}&\color{blue}{-7}&\color{blue}{3}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.