Divide using synthetic division $$ ( 6x^{3}+7x^{2}-x+26 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&6&7&-1&26\\& & 18& 75& \color{black}{222} \\ \hline &\color{blue}{6}&\color{blue}{25}&\color{blue}{74}&\color{orangered}{248} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+7x^{2}-x+26 }{ x-3 } = \color{blue}{6x^{2}+25x+74} ~+~ \frac{ \color{red}{ 248 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&7&-1&26\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 6 }&7&-1&26\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&7&-1&26\\& & \color{blue}{18} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 18 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}3&6&\color{orangered}{ 7 }&-1&26\\& & \color{orangered}{18} & & \\ \hline &6&\color{orangered}{25}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 25 } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&7&-1&26\\& & 18& \color{blue}{75} & \\ \hline &6&\color{blue}{25}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 75 } = \color{orangered}{ 74 } $
$$ \begin{array}{c|rrrr}3&6&7&\color{orangered}{ -1 }&26\\& & 18& \color{orangered}{75} & \\ \hline &6&25&\color{orangered}{74}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 74 } = \color{blue}{ 222 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&7&-1&26\\& & 18& 75& \color{blue}{222} \\ \hline &6&25&\color{blue}{74}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ 222 } = \color{orangered}{ 248 } $
$$ \begin{array}{c|rrrr}3&6&7&-1&\color{orangered}{ 26 }\\& & 18& 75& \color{orangered}{222} \\ \hline &\color{blue}{6}&\color{blue}{25}&\color{blue}{74}&\color{orangered}{248} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+25x+74 } $ with a remainder of $ \color{red}{ 248 } $.