Divide using synthetic division $$ ( 5x^{4}-20x-84 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&5&0&0&-20&-84\\& & -25& 125& -625& \color{black}{3225} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{125}&\color{blue}{-645}&\color{orangered}{3141} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-20x-84 }{ x+5 } = \color{blue}{5x^{3}-25x^{2}+125x-645} ~+~ \frac{ \color{red}{ 3141 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&5&0&0&-20&-84\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 5 }&0&0&-20&-84\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&5&0&0&-20&-84\\& & \color{blue}{-25} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrrr}-5&5&\color{orangered}{ 0 }&0&-20&-84\\& & \color{orangered}{-25} & & & \\ \hline &5&\color{orangered}{-25}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&5&0&0&-20&-84\\& & -25& \color{blue}{125} & & \\ \hline &5&\color{blue}{-25}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 125 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrrr}-5&5&0&\color{orangered}{ 0 }&-20&-84\\& & -25& \color{orangered}{125} & & \\ \hline &5&-25&\color{orangered}{125}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 125 } = \color{blue}{ -625 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&5&0&0&-20&-84\\& & -25& 125& \color{blue}{-625} & \\ \hline &5&-25&\color{blue}{125}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -625 \right) } = \color{orangered}{ -645 } $
$$ \begin{array}{c|rrrrr}-5&5&0&0&\color{orangered}{ -20 }&-84\\& & -25& 125& \color{orangered}{-625} & \\ \hline &5&-25&125&\color{orangered}{-645}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -645 \right) } = \color{blue}{ 3225 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&5&0&0&-20&-84\\& & -25& 125& -625& \color{blue}{3225} \\ \hline &5&-25&125&\color{blue}{-645}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -84 } + \color{orangered}{ 3225 } = \color{orangered}{ 3141 } $
$$ \begin{array}{c|rrrrr}-5&5&0&0&-20&\color{orangered}{ -84 }\\& & -25& 125& -625& \color{orangered}{3225} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{125}&\color{blue}{-645}&\color{orangered}{3141} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}-25x^{2}+125x-645 } $ with a remainder of $ \color{red}{ 3141 } $.