Divide using synthetic division $$ ( 2x^{3}+13x^{2}-22x+16 ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-8&2&13&-22&16\\& & -16& 24& \color{black}{-16} \\ \hline &\color{blue}{2}&\color{blue}{-3}&\color{blue}{2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+13x^{2}-22x+16 }{ x+8 } = \color{blue}{2x^{2}-3x+2} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&2&13&-22&16\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-8&\color{orangered}{ 2 }&13&-22&16\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 2 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&2&13&-22&16\\& & \color{blue}{-16} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-8&2&\color{orangered}{ 13 }&-22&16\\& & \color{orangered}{-16} & & \\ \hline &2&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&2&13&-22&16\\& & -16& \color{blue}{24} & \\ \hline &2&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 24 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-8&2&13&\color{orangered}{ -22 }&16\\& & -16& \color{orangered}{24} & \\ \hline &2&-3&\color{orangered}{2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 2 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&2&13&-22&16\\& & -16& 24& \color{blue}{-16} \\ \hline &2&-3&\color{blue}{2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-8&2&13&-22&\color{orangered}{ 16 }\\& & -16& 24& \color{orangered}{-16} \\ \hline &\color{blue}{2}&\color{blue}{-3}&\color{blue}{2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-3x+2 } $ with a remainder of $ \color{red}{ 0 } $.