Divide using synthetic division $$ ( 3x^{3}+2x^{2}-3x+54 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&3&2&-3&54\\& & -9& 21& \color{black}{-54} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{18}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+2x^{2}-3x+54 }{ x+3 } = \color{blue}{3x^{2}-7x+18} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&2&-3&54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 3 }&2&-3&54\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&2&-3&54\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-3&3&\color{orangered}{ 2 }&-3&54\\& & \color{orangered}{-9} & & \\ \hline &3&\color{orangered}{-7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&2&-3&54\\& & -9& \color{blue}{21} & \\ \hline &3&\color{blue}{-7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 21 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-3&3&2&\color{orangered}{ -3 }&54\\& & -9& \color{orangered}{21} & \\ \hline &3&-7&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 18 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&2&-3&54\\& & -9& 21& \color{blue}{-54} \\ \hline &3&-7&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 54 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-3&3&2&-3&\color{orangered}{ 54 }\\& & -9& 21& \color{orangered}{-54} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{18}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-7x+18 } $ with a remainder of $ \color{red}{ 0 } $.