Check if $ x-1.45 $ is a factor of $ 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&-10&8&1&0&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \frac{ 5539 }{ 2000 }& \frac{ 218631 }{ 40000 }& \color{black}{\frac{ 6340299 }{ 800000 }} \\ \hline &\color{blue}{4}&\color{blue}{-\frac{ 21 }{ 5 }}&\color{blue}{\frac{ 191 }{ 100 }}&\color{blue}{\frac{ 7539 }{ 2000 }}&\color{blue}{\frac{ 218631 }{ 40000 }}&\color{orangered}{-\frac{ 59701 }{ 800000 }} \end{array} $$Because the remainder $ \left( \color{red}{ -\frac{ 59701 }{ 800000 } } \right) $ is not zero, we conclude that the $ x-\frac{ 29 }{ 20 } $ is not a factor of $ 4x^{5}-10x^{4}+8x^{3}+x^{2}-8$.
Explanation
First we need to create a synthetic division table.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 29 }{ 20 } = 0 $ ( $ x = \color{blue}{ \frac{ 29 }{ 20 } } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&\color{orangered}{ 4 }&-10&8&1&0&-8\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 29 }{ 20 } } \cdot \color{blue}{ 4 } = \color{blue}{ \frac{ 29 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & \color{blue}{\frac{ 29 }{ 5 }} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \frac{ 29 }{ 5 } } = \color{orangered}{ -\frac{ 21 }{ 5 } } $
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&\color{orangered}{ -10 }&8&1&0&-8\\& & \color{orangered}{\frac{ 29 }{ 5 }} & & & & \\ \hline &4&\color{orangered}{-\frac{ 21 }{ 5 }}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 29 }{ 20 } } \cdot \color{blue}{ \left( -\frac{ 21 }{ 5 } \right) } = \color{blue}{ -\frac{ 609 }{ 100 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & \frac{ 29 }{ 5 }& \color{blue}{-\frac{ 609 }{ 100 }} & & & \\ \hline &4&\color{blue}{-\frac{ 21 }{ 5 }}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -\frac{ 609 }{ 100 } \right) } = \color{orangered}{ \frac{ 191 }{ 100 } } $
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&-10&\color{orangered}{ 8 }&1&0&-8\\& & \frac{ 29 }{ 5 }& \color{orangered}{-\frac{ 609 }{ 100 }} & & & \\ \hline &4&-\frac{ 21 }{ 5 }&\color{orangered}{\frac{ 191 }{ 100 }}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 29 }{ 20 } } \cdot \color{blue}{ \frac{ 191 }{ 100 } } = \color{blue}{ \frac{ 5539 }{ 2000 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \color{blue}{\frac{ 5539 }{ 2000 }} & & \\ \hline &4&-\frac{ 21 }{ 5 }&\color{blue}{\frac{ 191 }{ 100 }}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \frac{ 5539 }{ 2000 } } = \color{orangered}{ \frac{ 7539 }{ 2000 } } $
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&-10&8&\color{orangered}{ 1 }&0&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \color{orangered}{\frac{ 5539 }{ 2000 }} & & \\ \hline &4&-\frac{ 21 }{ 5 }&\frac{ 191 }{ 100 }&\color{orangered}{\frac{ 7539 }{ 2000 }}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 29 }{ 20 } } \cdot \color{blue}{ \frac{ 7539 }{ 2000 } } = \color{blue}{ \frac{ 218631 }{ 40000 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \frac{ 5539 }{ 2000 }& \color{blue}{\frac{ 218631 }{ 40000 }} & \\ \hline &4&-\frac{ 21 }{ 5 }&\frac{ 191 }{ 100 }&\color{blue}{\frac{ 7539 }{ 2000 }}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \frac{ 218631 }{ 40000 } } = \color{orangered}{ \frac{ 218631 }{ 40000 } } $
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&-10&8&1&\color{orangered}{ 0 }&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \frac{ 5539 }{ 2000 }& \color{orangered}{\frac{ 218631 }{ 40000 }} & \\ \hline &4&-\frac{ 21 }{ 5 }&\frac{ 191 }{ 100 }&\frac{ 7539 }{ 2000 }&\color{orangered}{\frac{ 218631 }{ 40000 }}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 29 }{ 20 } } \cdot \color{blue}{ \frac{ 218631 }{ 40000 } } = \color{blue}{ \frac{ 6340299 }{ 800000 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{\frac{ 29 }{ 20 }}&4&-10&8&1&0&-8\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \frac{ 5539 }{ 2000 }& \frac{ 218631 }{ 40000 }& \color{blue}{\frac{ 6340299 }{ 800000 }} \\ \hline &4&-\frac{ 21 }{ 5 }&\frac{ 191 }{ 100 }&\frac{ 7539 }{ 2000 }&\color{blue}{\frac{ 218631 }{ 40000 }}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \frac{ 6340299 }{ 800000 } } = \color{orangered}{ -\frac{ 59701 }{ 800000 } } $
$$ \begin{array}{c|rrrrrr}\frac{ 29 }{ 20 }&4&-10&8&1&0&\color{orangered}{ -8 }\\& & \frac{ 29 }{ 5 }& -\frac{ 609 }{ 100 }& \frac{ 5539 }{ 2000 }& \frac{ 218631 }{ 40000 }& \color{orangered}{\frac{ 6340299 }{ 800000 }} \\ \hline &\color{blue}{4}&\color{blue}{-\frac{ 21 }{ 5 }}&\color{blue}{\frac{ 191 }{ 100 }}&\color{blue}{\frac{ 7539 }{ 2000 }}&\color{blue}{\frac{ 218631 }{ 40000 }}&\color{orangered}{-\frac{ 59701 }{ 800000 }} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -\frac{ 59701 }{ 800000 } }\right)$.