Divide using synthetic division $$ ( x^{3}-3x^{2}-25x-21 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&1&-3&-25&-21\\& & 7& 28& \color{black}{21} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-3x^{2}-25x-21 }{ x-7 } = \color{blue}{x^{2}+4x+3} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-3&-25&-21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 1 }&-3&-25&-21\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-3&-25&-21\\& & \color{blue}{7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 7 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}7&1&\color{orangered}{ -3 }&-25&-21\\& & \color{orangered}{7} & & \\ \hline &1&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 4 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-3&-25&-21\\& & 7& \color{blue}{28} & \\ \hline &1&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 28 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}7&1&-3&\color{orangered}{ -25 }&-21\\& & 7& \color{orangered}{28} & \\ \hline &1&4&\color{orangered}{3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 3 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-3&-25&-21\\& & 7& 28& \color{blue}{21} \\ \hline &1&4&\color{blue}{3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 21 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}7&1&-3&-25&\color{orangered}{ -21 }\\& & 7& 28& \color{orangered}{21} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+4x+3 } $ with a remainder of $ \color{red}{ 0 } $.