Divide using synthetic division $$ ( 3x^{6}+5x^{4}-x+3 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}1&3&0&5&0&0&-1&3\\& & 3& 3& 8& 8& 8& \color{black}{7} \\ \hline &\color{blue}{3}&\color{blue}{3}&\color{blue}{8}&\color{blue}{8}&\color{blue}{8}&\color{blue}{7}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 3x^{6}+5x^{4}-x+3 }{ x-1 } = \color{blue}{3x^{5}+3x^{4}+8x^{3}+8x^{2}+8x+7} ~+~ \frac{ \color{red}{ 10 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}1&\color{orangered}{ 3 }&0&5&0&0&-1&3\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & \color{blue}{3} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 3 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}1&3&\color{orangered}{ 0 }&5&0&0&-1&3\\& & \color{orangered}{3} & & & & & \\ \hline &3&\color{orangered}{3}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & 3& \color{blue}{3} & & & & \\ \hline &3&\color{blue}{3}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 3 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}1&3&0&\color{orangered}{ 5 }&0&0&-1&3\\& & 3& \color{orangered}{3} & & & & \\ \hline &3&3&\color{orangered}{8}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 8 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & 3& 3& \color{blue}{8} & & & \\ \hline &3&3&\color{blue}{8}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}1&3&0&5&\color{orangered}{ 0 }&0&-1&3\\& & 3& 3& \color{orangered}{8} & & & \\ \hline &3&3&8&\color{orangered}{8}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 8 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & 3& 3& 8& \color{blue}{8} & & \\ \hline &3&3&8&\color{blue}{8}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}1&3&0&5&0&\color{orangered}{ 0 }&-1&3\\& & 3& 3& 8& \color{orangered}{8} & & \\ \hline &3&3&8&8&\color{orangered}{8}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 8 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & 3& 3& 8& 8& \color{blue}{8} & \\ \hline &3&3&8&8&\color{blue}{8}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 8 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrrr}1&3&0&5&0&0&\color{orangered}{ -1 }&3\\& & 3& 3& 8& 8& \color{orangered}{8} & \\ \hline &3&3&8&8&8&\color{orangered}{7}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 7 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&3&0&5&0&0&-1&3\\& & 3& 3& 8& 8& 8& \color{blue}{7} \\ \hline &3&3&8&8&8&\color{blue}{7}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 7 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrr}1&3&0&5&0&0&-1&\color{orangered}{ 3 }\\& & 3& 3& 8& 8& 8& \color{orangered}{7} \\ \hline &\color{blue}{3}&\color{blue}{3}&\color{blue}{8}&\color{blue}{8}&\color{blue}{8}&\color{blue}{7}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}+3x^{4}+8x^{3}+8x^{2}+8x+7 } $ with a remainder of $ \color{red}{ 10 } $.