Check if $ 3x-2 $ is a factor of $ 24x^{4}-10x^{3}-16x^{2}+17x-6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&24&-10&-16&17&-6\\& & 48& 76& 120& \color{black}{274} \\ \hline &\color{blue}{24}&\color{blue}{38}&\color{blue}{60}&\color{blue}{137}&\color{orangered}{268} \end{array} $$Because the remainder $ \left( \color{red}{ 268 } \right) $ is not zero, we conclude that the $ x-2 $ is not a factor of $ 24x^{4}-10x^{3}-16x^{2}+17x-6$.
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 -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&24&-10&-16&17&-6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 24 }&-10&-16&17&-6\\& & & & & \\ \hline &\color{orangered}{24}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 24 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&24&-10&-16&17&-6\\& & \color{blue}{48} & & & \\ \hline &\color{blue}{24}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 48 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrrr}2&24&\color{orangered}{ -10 }&-16&17&-6\\& & \color{orangered}{48} & & & \\ \hline &24&\color{orangered}{38}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 38 } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&24&-10&-16&17&-6\\& & 48& \color{blue}{76} & & \\ \hline &24&\color{blue}{38}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 76 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrrr}2&24&-10&\color{orangered}{ -16 }&17&-6\\& & 48& \color{orangered}{76} & & \\ \hline &24&38&\color{orangered}{60}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 60 } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&24&-10&-16&17&-6\\& & 48& 76& \color{blue}{120} & \\ \hline &24&38&\color{blue}{60}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 120 } = \color{orangered}{ 137 } $
$$ \begin{array}{c|rrrrr}2&24&-10&-16&\color{orangered}{ 17 }&-6\\& & 48& 76& \color{orangered}{120} & \\ \hline &24&38&60&\color{orangered}{137}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 137 } = \color{blue}{ 274 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&24&-10&-16&17&-6\\& & 48& 76& 120& \color{blue}{274} \\ \hline &24&38&60&\color{blue}{137}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 274 } = \color{orangered}{ 268 } $
$$ \begin{array}{c|rrrrr}2&24&-10&-16&17&\color{orangered}{ -6 }\\& & 48& 76& 120& \color{orangered}{274} \\ \hline &\color{blue}{24}&\color{blue}{38}&\color{blue}{60}&\color{blue}{137}&\color{orangered}{268} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 268 }\right)$.