Divide using synthetic division $$ ( 15x^{3}+14x^{2}-3x-2 ) \div ( x-0.4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 2 }{ 5 }&15&14&-3&-2\\& & 6& 8& \color{black}{2} \\ \hline &\color{blue}{15}&\color{blue}{20}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 15x^{3}+14x^{2}-3x-2 }{ x-\frac{ 2 }{ 5 } } = \color{blue}{15x^{2}+20x+5} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 2 }{ 5 } = 0 $ ( $ x = \color{blue}{ \frac{ 2 }{ 5 } } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 2 }{ 5 }}&15&14&-3&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 2 }{ 5 }&\color{orangered}{ 15 }&14&-3&-2\\& & & & \\ \hline &\color{orangered}{15}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 2 }{ 5 } } \cdot \color{blue}{ 15 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 2 }{ 5 }}&15&14&-3&-2\\& & \color{blue}{6} & & \\ \hline &\color{blue}{15}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 6 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}\frac{ 2 }{ 5 }&15&\color{orangered}{ 14 }&-3&-2\\& & \color{orangered}{6} & & \\ \hline &15&\color{orangered}{20}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 2 }{ 5 } } \cdot \color{blue}{ 20 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 2 }{ 5 }}&15&14&-3&-2\\& & 6& \color{blue}{8} & \\ \hline &15&\color{blue}{20}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 8 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}\frac{ 2 }{ 5 }&15&14&\color{orangered}{ -3 }&-2\\& & 6& \color{orangered}{8} & \\ \hline &15&20&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 2 }{ 5 } } \cdot \color{blue}{ 5 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 2 }{ 5 }}&15&14&-3&-2\\& & 6& 8& \color{blue}{2} \\ \hline &15&20&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 2 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}\frac{ 2 }{ 5 }&15&14&-3&\color{orangered}{ -2 }\\& & 6& 8& \color{orangered}{2} \\ \hline &\color{blue}{15}&\color{blue}{20}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x^{2}+20x+5 } $ with a remainder of $ \color{red}{ 0 } $.