Find remainder when $ -4x^{4}+12x^{3}+18x^{2}-14x+24 $ is divided by $ x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&-4&12&18&-14&24\\& & -16& -16& 8& \color{black}{-24} \\ \hline &\color{blue}{-4}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The remainder when $ -4x^{4}+12x^{3}+18x^{2}-14x+24 $ is divided by $ x-4 $ 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 -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&12&18&-14&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ -4 }&12&18&-14&24\\& & & & & \\ \hline &\color{orangered}{-4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&12&18&-14&24\\& & \color{blue}{-16} & & & \\ \hline &\color{blue}{-4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}4&-4&\color{orangered}{ 12 }&18&-14&24\\& & \color{orangered}{-16} & & & \\ \hline &-4&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&12&18&-14&24\\& & -16& \color{blue}{-16} & & \\ \hline &-4&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}4&-4&12&\color{orangered}{ 18 }&-14&24\\& & -16& \color{orangered}{-16} & & \\ \hline &-4&-4&\color{orangered}{2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 2 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&12&18&-14&24\\& & -16& -16& \color{blue}{8} & \\ \hline &-4&-4&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 8 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}4&-4&12&18&\color{orangered}{ -14 }&24\\& & -16& -16& \color{orangered}{8} & \\ \hline &-4&-4&2&\color{orangered}{-6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&12&18&-14&24\\& & -16& -16& 8& \color{blue}{-24} \\ \hline &-4&-4&2&\color{blue}{-6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&-4&12&18&-14&\color{orangered}{ 24 }\\& & -16& -16& 8& \color{orangered}{-24} \\ \hline &\color{blue}{-4}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.