Divide using synthetic division $$ ( 3x^{3}-24x^{2}-53x-66 ) \div ( x-10 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}10&3&-24&-53&-66\\& & 30& 60& \color{black}{70} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{4} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-24x^{2}-53x-66 }{ x-10 } = \color{blue}{3x^{2}+6x+7} ~+~ \frac{ \color{red}{ 4 } }{ x-10 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -10 = 0 $ ( $ x = \color{blue}{ 10 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{10}&3&-24&-53&-66\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}10&\color{orangered}{ 3 }&-24&-53&-66\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 3 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&3&-24&-53&-66\\& & \color{blue}{30} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 30 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}10&3&\color{orangered}{ -24 }&-53&-66\\& & \color{orangered}{30} & & \\ \hline &3&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 6 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&3&-24&-53&-66\\& & 30& \color{blue}{60} & \\ \hline &3&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -53 } + \color{orangered}{ 60 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}10&3&-24&\color{orangered}{ -53 }&-66\\& & 30& \color{orangered}{60} & \\ \hline &3&6&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 7 } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&3&-24&-53&-66\\& & 30& 60& \color{blue}{70} \\ \hline &3&6&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -66 } + \color{orangered}{ 70 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}10&3&-24&-53&\color{orangered}{ -66 }\\& & 30& 60& \color{orangered}{70} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+6x+7 } $ with a remainder of $ \color{red}{ 4 } $.