Divide using synthetic division $$ ( x^{4}-8x^{3}+48x^{2}-176x+260 ) \div ( -x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-8&48&-176&260\\& & -3& 33& -243& \color{black}{1257} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{81}&\color{blue}{-419}&\color{orangered}{1517} \end{array} $$The solution is:
$$ \frac{ x^{4}-8x^{3}+48x^{2}-176x+260 }{ x+3 } = \color{blue}{x^{3}-11x^{2}+81x-419} ~+~ \frac{ \color{red}{ 1517 } }{ x+3 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-8&48&-176&260\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-8&48&-176&260\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-8&48&-176&260\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -8 }&48&-176&260\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-8&48&-176&260\\& & -3& \color{blue}{33} & & \\ \hline &1&\color{blue}{-11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 48 } + \color{orangered}{ 33 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrr}-3&1&-8&\color{orangered}{ 48 }&-176&260\\& & -3& \color{orangered}{33} & & \\ \hline &1&-11&\color{orangered}{81}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 81 } = \color{blue}{ -243 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-8&48&-176&260\\& & -3& 33& \color{blue}{-243} & \\ \hline &1&-11&\color{blue}{81}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -176 } + \color{orangered}{ \left( -243 \right) } = \color{orangered}{ -419 } $
$$ \begin{array}{c|rrrrr}-3&1&-8&48&\color{orangered}{ -176 }&260\\& & -3& 33& \color{orangered}{-243} & \\ \hline &1&-11&81&\color{orangered}{-419}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -419 \right) } = \color{blue}{ 1257 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-8&48&-176&260\\& & -3& 33& -243& \color{blue}{1257} \\ \hline &1&-11&81&\color{blue}{-419}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 260 } + \color{orangered}{ 1257 } = \color{orangered}{ 1517 } $
$$ \begin{array}{c|rrrrr}-3&1&-8&48&-176&\color{orangered}{ 260 }\\& & -3& 33& -243& \color{orangered}{1257} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{81}&\color{blue}{-419}&\color{orangered}{1517} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-11x^{2}+81x-419 } $ with a remainder of $ \color{red}{ 1517 } $.