Divide using synthetic division $$ ( x^{3}-4.999x^{2}+6.996x-2.997 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & 1& -\frac{ 3999 }{ 1000 }& \color{black}{\frac{ 2997 }{ 1000 }} \\ \hline &\color{blue}{1}&\color{blue}{-\frac{ 3999 }{ 1000 }}&\color{blue}{\frac{ 2997 }{ 1000 }}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-\frac{ 4999 }{ 1000 }x^{2}+\frac{ 1749 }{ 250 }x-\frac{ 2997 }{ 1000 } }{ x-1 } = \color{blue}{x^{2}-\frac{ 3999 }{ 1000 }x+\frac{ 2997 }{ 1000 }} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 1 }&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & \color{blue}{1} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 4999 }{ 1000 } } + \color{orangered}{ 1 } = \color{orangered}{ -\frac{ 3999 }{ 1000 } } $
$$ \begin{array}{c|rrrr}1&1&\color{orangered}{ -\frac{ 4999 }{ 1000 } }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & \color{orangered}{1} & & \\ \hline &1&\color{orangered}{-\frac{ 3999 }{ 1000 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -\frac{ 3999 }{ 1000 } \right) } = \color{blue}{ -\frac{ 3999 }{ 1000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & 1& \color{blue}{-\frac{ 3999 }{ 1000 }} & \\ \hline &1&\color{blue}{-\frac{ 3999 }{ 1000 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ \frac{ 1749 }{ 250 } } + \color{orangered}{ \left( -\frac{ 3999 }{ 1000 } \right) } = \color{orangered}{ \frac{ 2997 }{ 1000 } } $
$$ \begin{array}{c|rrrr}1&1&-\frac{ 4999 }{ 1000 }&\color{orangered}{ \frac{ 1749 }{ 250 } }&-\frac{ 2997 }{ 1000 }\\& & 1& \color{orangered}{-\frac{ 3999 }{ 1000 }} & \\ \hline &1&-\frac{ 3999 }{ 1000 }&\color{orangered}{\frac{ 2997 }{ 1000 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \frac{ 2997 }{ 1000 } } = \color{blue}{ \frac{ 2997 }{ 1000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&-\frac{ 2997 }{ 1000 }\\& & 1& -\frac{ 3999 }{ 1000 }& \color{blue}{\frac{ 2997 }{ 1000 }} \\ \hline &1&-\frac{ 3999 }{ 1000 }&\color{blue}{\frac{ 2997 }{ 1000 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -\frac{ 2997 }{ 1000 } } + \color{orangered}{ \frac{ 2997 }{ 1000 } } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}1&1&-\frac{ 4999 }{ 1000 }&\frac{ 1749 }{ 250 }&\color{orangered}{ -\frac{ 2997 }{ 1000 } }\\& & 1& -\frac{ 3999 }{ 1000 }& \color{orangered}{\frac{ 2997 }{ 1000 }} \\ \hline &\color{blue}{1}&\color{blue}{-\frac{ 3999 }{ 1000 }}&\color{blue}{\frac{ 2997 }{ 1000 }}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-\frac{ 3999 }{ 1000 }x+\frac{ 2997 }{ 1000 } } $ with a remainder of $ \color{red}{ 0 } $.