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