Divide using synthetic division $$ ( 2x^{3}+x^{2}-22x+20 ) \div ( x-1.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 2 }&2&1&-22&20\\& & 3& 6& \color{black}{-24} \\ \hline &\color{blue}{2}&\color{blue}{4}&\color{blue}{-16}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+x^{2}-22x+20 }{ x-\frac{ 3 }{ 2 } } = \color{blue}{2x^{2}+4x-16} \color{red}{~-~} \frac{ \color{red}{ 4 } }{ x-\frac{ 3 }{ 2 } } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 3 }{ 2 } = 0 $ ( $ x = \color{blue}{ \frac{ 3 }{ 2 } } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 2 }}&2&1&-22&20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 2 }&\color{orangered}{ 2 }&1&-22&20\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 2 } } \cdot \color{blue}{ 2 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 2 }}&2&1&-22&20\\& & \color{blue}{3} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 3 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 2 }&2&\color{orangered}{ 1 }&-22&20\\& & \color{orangered}{3} & & \\ \hline &2&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 2 } } \cdot \color{blue}{ 4 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 2 }}&2&1&-22&20\\& & 3& \color{blue}{6} & \\ \hline &2&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 6 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 2 }&2&1&\color{orangered}{ -22 }&20\\& & 3& \color{orangered}{6} & \\ \hline &2&4&\color{orangered}{-16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 2 } } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 2 }}&2&1&-22&20\\& & 3& 6& \color{blue}{-24} \\ \hline &2&4&\color{blue}{-16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 2 }&2&1&-22&\color{orangered}{ 20 }\\& & 3& 6& \color{orangered}{-24} \\ \hline &\color{blue}{2}&\color{blue}{4}&\color{blue}{-16}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+4x-16 } $ with a remainder of $ \color{red}{ -4 } $.