Divide using synthetic division $$ ( 4x^{4}+19x^{3}-73x^{2}-64x+41 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&4&19&-73&-64&41\\& & -28& 63& 70& \color{black}{-42} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{-10}&\color{blue}{6}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+19x^{3}-73x^{2}-64x+41 }{ x+7 } = \color{blue}{4x^{3}-9x^{2}-10x+6} \color{red}{~-~} \frac{ \color{red}{ 1 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&4&19&-73&-64&41\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 4 }&19&-73&-64&41\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 4 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&4&19&-73&-64&41\\& & \color{blue}{-28} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-7&4&\color{orangered}{ 19 }&-73&-64&41\\& & \color{orangered}{-28} & & & \\ \hline &4&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&4&19&-73&-64&41\\& & -28& \color{blue}{63} & & \\ \hline &4&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -73 } + \color{orangered}{ 63 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}-7&4&19&\color{orangered}{ -73 }&-64&41\\& & -28& \color{orangered}{63} & & \\ \hline &4&-9&\color{orangered}{-10}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&4&19&-73&-64&41\\& & -28& 63& \color{blue}{70} & \\ \hline &4&-9&\color{blue}{-10}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -64 } + \color{orangered}{ 70 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}-7&4&19&-73&\color{orangered}{ -64 }&41\\& & -28& 63& \color{orangered}{70} & \\ \hline &4&-9&-10&\color{orangered}{6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 6 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&4&19&-73&-64&41\\& & -28& 63& 70& \color{blue}{-42} \\ \hline &4&-9&-10&\color{blue}{6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 41 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-7&4&19&-73&-64&\color{orangered}{ 41 }\\& & -28& 63& 70& \color{orangered}{-42} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{-10}&\color{blue}{6}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-9x^{2}-10x+6 } $ with a remainder of $ \color{red}{ -1 } $.