Divide using synthetic division $$ ( x^{3}-13x^{2}+44x-18 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&1&-13&44&-18\\& & 4& -36& \color{black}{32} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{8}&\color{orangered}{14} \end{array} $$The solution is:
$$ \frac{ x^{3}-13x^{2}+44x-18 }{ x-4 } = \color{blue}{x^{2}-9x+8} ~+~ \frac{ \color{red}{ 14 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-13&44&-18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 1 }&-13&44&-18\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-13&44&-18\\& & \color{blue}{4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 4 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}4&1&\color{orangered}{ -13 }&44&-18\\& & \color{orangered}{4} & & \\ \hline &1&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-13&44&-18\\& & 4& \color{blue}{-36} & \\ \hline &1&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 44 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}4&1&-13&\color{orangered}{ 44 }&-18\\& & 4& \color{orangered}{-36} & \\ \hline &1&-9&\color{orangered}{8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-13&44&-18\\& & 4& -36& \color{blue}{32} \\ \hline &1&-9&\color{blue}{8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 32 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}4&1&-13&44&\color{orangered}{ -18 }\\& & 4& -36& \color{orangered}{32} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{8}&\color{orangered}{14} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-9x+8 } $ with a remainder of $ \color{red}{ 14 } $.