Find remainder when $ 8x^{2}-7x+10 $ is divided by $ 8x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&8&-7&10\\& & -8& \color{black}{15} \\ \hline &\color{blue}{8}&\color{blue}{-15}&\color{orangered}{25} \end{array} $$The remainder when $ 8x^{2}-7x+10 $ is divided by $ x+1 $ is $ \, \color{red}{ 25 } $.
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.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&-7&10\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 8 }&-7&10\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 8 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&-7&10\\& & \color{blue}{-8} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrr}-1&8&\color{orangered}{ -7 }&10\\& & \color{orangered}{-8} & \\ \hline &8&\color{orangered}{-15}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&-7&10\\& & -8& \color{blue}{15} \\ \hline &8&\color{blue}{-15}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 15 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrr}-1&8&-7&\color{orangered}{ 10 }\\& & -8& \color{orangered}{15} \\ \hline &\color{blue}{8}&\color{blue}{-15}&\color{orangered}{25} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 25 }\right) $.