Divide using synthetic division $$ ( x^{3}-3x^{2}-76x+288 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&1&-3&-76&288\\& & 4& 4& \color{black}{-288} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-72}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-3x^{2}-76x+288 }{ x-4 } = \color{blue}{x^{2}+x-72} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-3&-76&288\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 1 }&-3&-76&288\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-3&-76&288\\& & \color{blue}{4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 4 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}4&1&\color{orangered}{ -3 }&-76&288\\& & \color{orangered}{4} & & \\ \hline &1&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-3&-76&288\\& & 4& \color{blue}{4} & \\ \hline &1&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -76 } + \color{orangered}{ 4 } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrr}4&1&-3&\color{orangered}{ -76 }&288\\& & 4& \color{orangered}{4} & \\ \hline &1&1&\color{orangered}{-72}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -72 \right) } = \color{blue}{ -288 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-3&-76&288\\& & 4& 4& \color{blue}{-288} \\ \hline &1&1&\color{blue}{-72}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 288 } + \color{orangered}{ \left( -288 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}4&1&-3&-76&\color{orangered}{ 288 }\\& & 4& 4& \color{orangered}{-288} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-72}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+x-72 } $ with a remainder of $ \color{red}{ 0 } $.