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