Divide using synthetic division $$ ( 5x^{4}-8x^{3}+10x-22 ) \div ( 5x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&5&-8&0&10&-22\\& & 40& 256& 2048& \color{black}{16464} \\ \hline &\color{blue}{5}&\color{blue}{32}&\color{blue}{256}&\color{blue}{2058}&\color{orangered}{16442} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-8x^{3}+10x-22 }{ x-8 } = \color{blue}{5x^{3}+32x^{2}+256x+2058} ~+~ \frac{ \color{red}{ 16442 } }{ x-8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&5&-8&0&10&-22\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 5 }&-8&0&10&-22\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 5 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&5&-8&0&10&-22\\& & \color{blue}{40} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 40 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrr}8&5&\color{orangered}{ -8 }&0&10&-22\\& & \color{orangered}{40} & & & \\ \hline &5&\color{orangered}{32}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 32 } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&5&-8&0&10&-22\\& & 40& \color{blue}{256} & & \\ \hline &5&\color{blue}{32}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 256 } = \color{orangered}{ 256 } $
$$ \begin{array}{c|rrrrr}8&5&-8&\color{orangered}{ 0 }&10&-22\\& & 40& \color{orangered}{256} & & \\ \hline &5&32&\color{orangered}{256}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 256 } = \color{blue}{ 2048 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&5&-8&0&10&-22\\& & 40& 256& \color{blue}{2048} & \\ \hline &5&32&\color{blue}{256}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 2048 } = \color{orangered}{ 2058 } $
$$ \begin{array}{c|rrrrr}8&5&-8&0&\color{orangered}{ 10 }&-22\\& & 40& 256& \color{orangered}{2048} & \\ \hline &5&32&256&\color{orangered}{2058}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 2058 } = \color{blue}{ 16464 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&5&-8&0&10&-22\\& & 40& 256& 2048& \color{blue}{16464} \\ \hline &5&32&256&\color{blue}{2058}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 16464 } = \color{orangered}{ 16442 } $
$$ \begin{array}{c|rrrrr}8&5&-8&0&10&\color{orangered}{ -22 }\\& & 40& 256& 2048& \color{orangered}{16464} \\ \hline &\color{blue}{5}&\color{blue}{32}&\color{blue}{256}&\color{blue}{2058}&\color{orangered}{16442} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}+32x^{2}+256x+2058 } $ with a remainder of $ \color{red}{ 16442 } $.