Divide using synthetic division $$ ( 3x^{3}-16x^{2}+17x-4 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&3&-16&17&-4\\& & 3& -13& \color{black}{4} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{blue}{4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-16x^{2}+17x-4 }{ x-1 } = \color{blue}{3x^{2}-13x+4} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&-16&17&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 3 }&-16&17&-4\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&-16&17&-4\\& & \color{blue}{3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 3 } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}1&3&\color{orangered}{ -16 }&17&-4\\& & \color{orangered}{3} & & \\ \hline &3&\color{orangered}{-13}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&-16&17&-4\\& & 3& \color{blue}{-13} & \\ \hline &3&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -13 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}1&3&-16&\color{orangered}{ 17 }&-4\\& & 3& \color{orangered}{-13} & \\ \hline &3&-13&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 4 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&-16&17&-4\\& & 3& -13& \color{blue}{4} \\ \hline &3&-13&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}1&3&-16&17&\color{orangered}{ -4 }\\& & 3& -13& \color{orangered}{4} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{blue}{4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-13x+4 } $ with a remainder of $ \color{red}{ 0 } $.