Find remainder when $ 32x^{6}-36x^{5}-13x^{3}+34x^{2}+28 $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}1&32&-36&0&-13&34&0&28\\& & 32& -4& -4& -17& 17& \color{black}{17} \\ \hline &\color{blue}{32}&\color{blue}{-4}&\color{blue}{-4}&\color{blue}{-17}&\color{blue}{17}&\color{blue}{17}&\color{orangered}{45} \end{array} $$The remainder when $ 32x^{6}-36x^{5}-13x^{3}+34x^{2}+28 $ is divided by $ x-1 $ is $ \, \color{red}{ 45 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}1&\color{orangered}{ 32 }&-36&0&-13&34&0&28\\& & & & & & & \\ \hline &\color{orangered}{32}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 32 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & \color{blue}{32} & & & & & \\ \hline &\color{blue}{32}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 32 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrr}1&32&\color{orangered}{ -36 }&0&-13&34&0&28\\& & \color{orangered}{32} & & & & & \\ \hline &32&\color{orangered}{-4}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & 32& \color{blue}{-4} & & & & \\ \hline &32&\color{blue}{-4}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrr}1&32&-36&\color{orangered}{ 0 }&-13&34&0&28\\& & 32& \color{orangered}{-4} & & & & \\ \hline &32&-4&\color{orangered}{-4}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & 32& -4& \color{blue}{-4} & & & \\ \hline &32&-4&\color{blue}{-4}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrrr}1&32&-36&0&\color{orangered}{ -13 }&34&0&28\\& & 32& -4& \color{orangered}{-4} & & & \\ \hline &32&-4&-4&\color{orangered}{-17}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -17 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & 32& -4& -4& \color{blue}{-17} & & \\ \hline &32&-4&-4&\color{blue}{-17}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ \left( -17 \right) } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrrrr}1&32&-36&0&-13&\color{orangered}{ 34 }&0&28\\& & 32& -4& -4& \color{orangered}{-17} & & \\ \hline &32&-4&-4&-17&\color{orangered}{17}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 17 } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & 32& -4& -4& -17& \color{blue}{17} & \\ \hline &32&-4&-4&-17&\color{blue}{17}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 17 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrrrr}1&32&-36&0&-13&34&\color{orangered}{ 0 }&28\\& & 32& -4& -4& -17& \color{orangered}{17} & \\ \hline &32&-4&-4&-17&17&\color{orangered}{17}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 17 } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&32&-36&0&-13&34&0&28\\& & 32& -4& -4& -17& 17& \color{blue}{17} \\ \hline &32&-4&-4&-17&17&\color{blue}{17}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 28 } + \color{orangered}{ 17 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrrrrr}1&32&-36&0&-13&34&0&\color{orangered}{ 28 }\\& & 32& -4& -4& -17& 17& \color{orangered}{17} \\ \hline &\color{blue}{32}&\color{blue}{-4}&\color{blue}{-4}&\color{blue}{-17}&\color{blue}{17}&\color{blue}{17}&\color{orangered}{45} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 45 }\right) $.