Divide using synthetic division $$ ( 2x^{3}-15x^{2}+34x-21 ) \div ( x-1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&2&-15&34&-21\\& & 2& -13& \color{black}{21} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{21}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 2x^{3}-15x^{2}+34x-21 }{ x-1 } = \color{blue}{2x^{2}-13x+21} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-15&34&-21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 2 }&-15&34&-21\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-15&34&-21\\& & \color{blue}{2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 2 } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}1&2&\color{orangered}{ -15 }&34&-21\\& & \color{orangered}{2} & & \\ \hline &2&\color{orangered}{-13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ -13 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-15&34&-21\\& & 2& \color{blue}{-13} & \\ \hline &2&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ \left( -13 \right) } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}1&2&-15&\color{orangered}{ 34 }&-21\\& & 2& \color{orangered}{-13} & \\ \hline &2&-13&\color{orangered}{21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 21 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-15&34&-21\\& & 2& -13& \color{blue}{21} \\ \hline &2&-13&\color{blue}{21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 21 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}1&2&-15&34&\color{orangered}{ -21 }\\& & 2& -13& \color{orangered}{21} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{21}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-13x+21 } $ with a remainder of $ \color{red}{ 0 } $.