Divide using synthetic division $$ ( 4x^{3}-28x^{2}+21x+18 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&4&-28&21&18\\& & 24& -24& \color{black}{-18} \\ \hline &\color{blue}{4}&\color{blue}{-4}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-28x^{2}+21x+18 }{ x-6 } = \color{blue}{4x^{2}-4x-3} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-28&21&18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 4 }&-28&21&18\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-28&21&18\\& & \color{blue}{24} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 24 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}6&4&\color{orangered}{ -28 }&21&18\\& & \color{orangered}{24} & & \\ \hline &4&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-28&21&18\\& & 24& \color{blue}{-24} & \\ \hline &4&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}6&4&-28&\color{orangered}{ 21 }&18\\& & 24& \color{orangered}{-24} & \\ \hline &4&-4&\color{orangered}{-3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-28&21&18\\& & 24& -24& \color{blue}{-18} \\ \hline &4&-4&\color{blue}{-3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}6&4&-28&21&\color{orangered}{ 18 }\\& & 24& -24& \color{orangered}{-18} \\ \hline &\color{blue}{4}&\color{blue}{-4}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-4x-3 } $ with a remainder of $ \color{red}{ 0 } $.