Divide using synthetic division $$ ( x^{3}-13x^{2}+54x-72 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&1&-13&54&-72\\& & 3& -30& \color{black}{72} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{24}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-13x^{2}+54x-72 }{ x-3 } = \color{blue}{x^{2}-10x+24} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-13&54&-72\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 1 }&-13&54&-72\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-13&54&-72\\& & \color{blue}{3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 3 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}3&1&\color{orangered}{ -13 }&54&-72\\& & \color{orangered}{3} & & \\ \hline &1&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-13&54&-72\\& & 3& \color{blue}{-30} & \\ \hline &1&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 54 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}3&1&-13&\color{orangered}{ 54 }&-72\\& & 3& \color{orangered}{-30} & \\ \hline &1&-10&\color{orangered}{24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 24 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-13&54&-72\\& & 3& -30& \color{blue}{72} \\ \hline &1&-10&\color{blue}{24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -72 } + \color{orangered}{ 72 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}3&1&-13&54&\color{orangered}{ -72 }\\& & 3& -30& \color{orangered}{72} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{24}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-10x+24 } $ with a remainder of $ \color{red}{ 0 } $.