Divide using synthetic division $$ ( x^{4}+8x^{3}+13x^{2}-30x-72 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&1&8&13&-30&-72\\& & 3& 33& 138& \color{black}{324} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{46}&\color{blue}{108}&\color{orangered}{252} \end{array} $$The solution is:
$$ \frac{ x^{4}+8x^{3}+13x^{2}-30x-72 }{ x-3 } = \color{blue}{x^{3}+11x^{2}+46x+108} ~+~ \frac{ \color{red}{ 252 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&8&13&-30&-72\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 1 }&8&13&-30&-72\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&8&13&-30&-72\\& & \color{blue}{3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 3 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrr}3&1&\color{orangered}{ 8 }&13&-30&-72\\& & \color{orangered}{3} & & & \\ \hline &1&\color{orangered}{11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 11 } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&8&13&-30&-72\\& & 3& \color{blue}{33} & & \\ \hline &1&\color{blue}{11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 33 } = \color{orangered}{ 46 } $
$$ \begin{array}{c|rrrrr}3&1&8&\color{orangered}{ 13 }&-30&-72\\& & 3& \color{orangered}{33} & & \\ \hline &1&11&\color{orangered}{46}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 46 } = \color{blue}{ 138 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&8&13&-30&-72\\& & 3& 33& \color{blue}{138} & \\ \hline &1&11&\color{blue}{46}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 138 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrrr}3&1&8&13&\color{orangered}{ -30 }&-72\\& & 3& 33& \color{orangered}{138} & \\ \hline &1&11&46&\color{orangered}{108}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 108 } = \color{blue}{ 324 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&8&13&-30&-72\\& & 3& 33& 138& \color{blue}{324} \\ \hline &1&11&46&\color{blue}{108}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -72 } + \color{orangered}{ 324 } = \color{orangered}{ 252 } $
$$ \begin{array}{c|rrrrr}3&1&8&13&-30&\color{orangered}{ -72 }\\& & 3& 33& 138& \color{orangered}{324} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{46}&\color{blue}{108}&\color{orangered}{252} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+11x^{2}+46x+108 } $ with a remainder of $ \color{red}{ 252 } $.