Divide using synthetic division $$ ( -4x^{4}-3x^{2}+4x-25 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&-4&0&-3&4&-25\\& & -16& -64& -268& \color{black}{-1056} \\ \hline &\color{blue}{-4}&\color{blue}{-16}&\color{blue}{-67}&\color{blue}{-264}&\color{orangered}{-1081} \end{array} $$The solution is:
$$ \frac{ -4x^{4}-3x^{2}+4x-25 }{ x-4 } = \color{blue}{-4x^{3}-16x^{2}-67x-264} \color{red}{~-~} \frac{ \color{red}{ 1081 } }{ x-4 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&0&-3&4&-25\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ -4 }&0&-3&4&-25\\& & & & & \\ \hline &\color{orangered}{-4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&0&-3&4&-25\\& & \color{blue}{-16} & & & \\ \hline &\color{blue}{-4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}4&-4&\color{orangered}{ 0 }&-3&4&-25\\& & \color{orangered}{-16} & & & \\ \hline &-4&\color{orangered}{-16}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&0&-3&4&-25\\& & -16& \color{blue}{-64} & & \\ \hline &-4&\color{blue}{-16}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -67 } $
$$ \begin{array}{c|rrrrr}4&-4&0&\color{orangered}{ -3 }&4&-25\\& & -16& \color{orangered}{-64} & & \\ \hline &-4&-16&\color{orangered}{-67}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -67 \right) } = \color{blue}{ -268 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&0&-3&4&-25\\& & -16& -64& \color{blue}{-268} & \\ \hline &-4&-16&\color{blue}{-67}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -268 \right) } = \color{orangered}{ -264 } $
$$ \begin{array}{c|rrrrr}4&-4&0&-3&\color{orangered}{ 4 }&-25\\& & -16& -64& \color{orangered}{-268} & \\ \hline &-4&-16&-67&\color{orangered}{-264}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -264 \right) } = \color{blue}{ -1056 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-4&0&-3&4&-25\\& & -16& -64& -268& \color{blue}{-1056} \\ \hline &-4&-16&-67&\color{blue}{-264}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -1056 \right) } = \color{orangered}{ -1081 } $
$$ \begin{array}{c|rrrrr}4&-4&0&-3&4&\color{orangered}{ -25 }\\& & -16& -64& -268& \color{orangered}{-1056} \\ \hline &\color{blue}{-4}&\color{blue}{-16}&\color{blue}{-67}&\color{blue}{-264}&\color{orangered}{-1081} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{3}-16x^{2}-67x-264 } $ with a remainder of $ \color{red}{ -1081 } $.