Divide using synthetic division $$ ( 3x^{4}-11x^{3}+13x^{2}-4 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&3&-11&13&0&-4\\& & 3& -8& 5& \color{black}{5} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{5}&\color{blue}{5}&\color{orangered}{1} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-11x^{3}+13x^{2}-4 }{ x-1 } = \color{blue}{3x^{3}-8x^{2}+5x+5} ~+~ \frac{ \color{red}{ 1 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&13&0&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 3 }&-11&13&0&-4\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&13&0&-4\\& & \color{blue}{3} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 3 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}1&3&\color{orangered}{ -11 }&13&0&-4\\& & \color{orangered}{3} & & & \\ \hline &3&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&13&0&-4\\& & 3& \color{blue}{-8} & & \\ \hline &3&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}1&3&-11&\color{orangered}{ 13 }&0&-4\\& & 3& \color{orangered}{-8} & & \\ \hline &3&-8&\color{orangered}{5}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 5 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&13&0&-4\\& & 3& -8& \color{blue}{5} & \\ \hline &3&-8&\color{blue}{5}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}1&3&-11&13&\color{orangered}{ 0 }&-4\\& & 3& -8& \color{orangered}{5} & \\ \hline &3&-8&5&\color{orangered}{5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 5 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&13&0&-4\\& & 3& -8& 5& \color{blue}{5} \\ \hline &3&-8&5&\color{blue}{5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 5 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}1&3&-11&13&0&\color{orangered}{ -4 }\\& & 3& -8& 5& \color{orangered}{5} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{5}&\color{blue}{5}&\color{orangered}{1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-8x^{2}+5x+5 } $ with a remainder of $ \color{red}{ 1 } $.