Divide using synthetic division $$ ( 5x^{3}-6x+1 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&5&0&-6&1\\& & 45& 405& \color{black}{3591} \\ \hline &\color{blue}{5}&\color{blue}{45}&\color{blue}{399}&\color{orangered}{3592} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-6x+1 }{ x-9 } = \color{blue}{5x^{2}+45x+399} ~+~ \frac{ \color{red}{ 3592 } }{ x-9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{9}&5&0&-6&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 5 }&0&-6&1\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 5 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&5&0&-6&1\\& & \color{blue}{45} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 45 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrr}9&5&\color{orangered}{ 0 }&-6&1\\& & \color{orangered}{45} & & \\ \hline &5&\color{orangered}{45}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 45 } = \color{blue}{ 405 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&5&0&-6&1\\& & 45& \color{blue}{405} & \\ \hline &5&\color{blue}{45}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 405 } = \color{orangered}{ 399 } $
$$ \begin{array}{c|rrrr}9&5&0&\color{orangered}{ -6 }&1\\& & 45& \color{orangered}{405} & \\ \hline &5&45&\color{orangered}{399}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 399 } = \color{blue}{ 3591 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&5&0&-6&1\\& & 45& 405& \color{blue}{3591} \\ \hline &5&45&\color{blue}{399}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 3591 } = \color{orangered}{ 3592 } $
$$ \begin{array}{c|rrrr}9&5&0&-6&\color{orangered}{ 1 }\\& & 45& 405& \color{orangered}{3591} \\ \hline &\color{blue}{5}&\color{blue}{45}&\color{blue}{399}&\color{orangered}{3592} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+45x+399 } $ with a remainder of $ \color{red}{ 3592 } $.