Divide using synthetic division $$ ( x^{4}-10x^{3}+37x^{2}-60x+36 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&1&-10&37&-60&36\\& & 2& -16& 42& \color{black}{-36} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{21}&\color{blue}{-18}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-10x^{3}+37x^{2}-60x+36 }{ x-2 } = \color{blue}{x^{3}-8x^{2}+21x-18} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-10&37&-60&36\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 1 }&-10&37&-60&36\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-10&37&-60&36\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 2 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}2&1&\color{orangered}{ -10 }&37&-60&36\\& & \color{orangered}{2} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-10&37&-60&36\\& & 2& \color{blue}{-16} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}2&1&-10&\color{orangered}{ 37 }&-60&36\\& & 2& \color{orangered}{-16} & & \\ \hline &1&-8&\color{orangered}{21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 21 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-10&37&-60&36\\& & 2& -16& \color{blue}{42} & \\ \hline &1&-8&\color{blue}{21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ 42 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}2&1&-10&37&\color{orangered}{ -60 }&36\\& & 2& -16& \color{orangered}{42} & \\ \hline &1&-8&21&\color{orangered}{-18}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-10&37&-60&36\\& & 2& -16& 42& \color{blue}{-36} \\ \hline &1&-8&21&\color{blue}{-18}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}2&1&-10&37&-60&\color{orangered}{ 36 }\\& & 2& -16& 42& \color{orangered}{-36} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{21}&\color{blue}{-18}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-8x^{2}+21x-18 } $ with a remainder of $ \color{red}{ 0 } $.