Divide using synthetic division $$ ( x^{4}+3x^{3}-9x^{2}+15x-10 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&1&3&-9&15&-10\\& & 1& 4& -5& \color{black}{10} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-5}&\color{blue}{10}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+3x^{3}-9x^{2}+15x-10 }{ x-1 } = \color{blue}{x^{3}+4x^{2}-5x+10} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&3&-9&15&-10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 1 }&3&-9&15&-10\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&3&-9&15&-10\\& & \color{blue}{1} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 1 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}1&1&\color{orangered}{ 3 }&-9&15&-10\\& & \color{orangered}{1} & & & \\ \hline &1&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 4 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&3&-9&15&-10\\& & 1& \color{blue}{4} & & \\ \hline &1&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 4 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}1&1&3&\color{orangered}{ -9 }&15&-10\\& & 1& \color{orangered}{4} & & \\ \hline &1&4&\color{orangered}{-5}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&3&-9&15&-10\\& & 1& 4& \color{blue}{-5} & \\ \hline &1&4&\color{blue}{-5}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrr}1&1&3&-9&\color{orangered}{ 15 }&-10\\& & 1& 4& \color{orangered}{-5} & \\ \hline &1&4&-5&\color{orangered}{10}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 10 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&3&-9&15&-10\\& & 1& 4& -5& \color{blue}{10} \\ \hline &1&4&-5&\color{blue}{10}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 10 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}1&1&3&-9&15&\color{orangered}{ -10 }\\& & 1& 4& -5& \color{orangered}{10} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-5}&\color{blue}{10}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+4x^{2}-5x+10 } $ with a remainder of $ \color{red}{ 0 } $.