Find remainder when $ 4x^{5}-19x^{4}+10x^{3}-9x^{2}+5x+5 $ is divided by $ x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&4&-19&10&-9&5&5\\& & 16& -12& -8& -68& \color{black}{-252} \\ \hline &\color{blue}{4}&\color{blue}{-3}&\color{blue}{-2}&\color{blue}{-17}&\color{blue}{-63}&\color{orangered}{-247} \end{array} $$The remainder when $ 4x^{5}-19x^{4}+10x^{3}-9x^{2}+5x+5 $ is divided by $ x-4 $ is $ \, \color{red}{ -247 } $.
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 4 }&-19&10&-9&5&5\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & \color{blue}{16} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 16 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}4&4&\color{orangered}{ -19 }&10&-9&5&5\\& & \color{orangered}{16} & & & & \\ \hline &4&\color{orangered}{-3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & 16& \color{blue}{-12} & & & \\ \hline &4&\color{blue}{-3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}4&4&-19&\color{orangered}{ 10 }&-9&5&5\\& & 16& \color{orangered}{-12} & & & \\ \hline &4&-3&\color{orangered}{-2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & 16& -12& \color{blue}{-8} & & \\ \hline &4&-3&\color{blue}{-2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrr}4&4&-19&10&\color{orangered}{ -9 }&5&5\\& & 16& -12& \color{orangered}{-8} & & \\ \hline &4&-3&-2&\color{orangered}{-17}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -68 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & 16& -12& -8& \color{blue}{-68} & \\ \hline &4&-3&-2&\color{blue}{-17}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -63 } $
$$ \begin{array}{c|rrrrrr}4&4&-19&10&-9&\color{orangered}{ 5 }&5\\& & 16& -12& -8& \color{orangered}{-68} & \\ \hline &4&-3&-2&-17&\color{orangered}{-63}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -63 \right) } = \color{blue}{ -252 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-19&10&-9&5&5\\& & 16& -12& -8& -68& \color{blue}{-252} \\ \hline &4&-3&-2&-17&\color{blue}{-63}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -252 \right) } = \color{orangered}{ -247 } $
$$ \begin{array}{c|rrrrrr}4&4&-19&10&-9&5&\color{orangered}{ 5 }\\& & 16& -12& -8& -68& \color{orangered}{-252} \\ \hline &\color{blue}{4}&\color{blue}{-3}&\color{blue}{-2}&\color{blue}{-17}&\color{blue}{-63}&\color{orangered}{-247} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -247 }\right) $.