Divide using synthetic division $$ ( 4x^{5}+2x^{4}-8x+3 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&4&2&0&0&-8&3\\& & -16& 56& -224& 896& \color{black}{-3552} \\ \hline &\color{blue}{4}&\color{blue}{-14}&\color{blue}{56}&\color{blue}{-224}&\color{blue}{888}&\color{orangered}{-3549} \end{array} $$The solution is:
$$ \frac{ 4x^{5}+2x^{4}-8x+3 }{ x+4 } = \color{blue}{4x^{4}-14x^{3}+56x^{2}-224x+888} \color{red}{~-~} \frac{ \color{red}{ 3549 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 4 }&2&0&0&-8&3\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & \color{blue}{-16} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrrrr}-4&4&\color{orangered}{ 2 }&0&0&-8&3\\& & \color{orangered}{-16} & & & & \\ \hline &4&\color{orangered}{-14}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 56 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & -16& \color{blue}{56} & & & \\ \hline &4&\color{blue}{-14}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 56 } = \color{orangered}{ 56 } $
$$ \begin{array}{c|rrrrrr}-4&4&2&\color{orangered}{ 0 }&0&-8&3\\& & -16& \color{orangered}{56} & & & \\ \hline &4&-14&\color{orangered}{56}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 56 } = \color{blue}{ -224 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & -16& 56& \color{blue}{-224} & & \\ \hline &4&-14&\color{blue}{56}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -224 \right) } = \color{orangered}{ -224 } $
$$ \begin{array}{c|rrrrrr}-4&4&2&0&\color{orangered}{ 0 }&-8&3\\& & -16& 56& \color{orangered}{-224} & & \\ \hline &4&-14&56&\color{orangered}{-224}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -224 \right) } = \color{blue}{ 896 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & -16& 56& -224& \color{blue}{896} & \\ \hline &4&-14&56&\color{blue}{-224}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 896 } = \color{orangered}{ 888 } $
$$ \begin{array}{c|rrrrrr}-4&4&2&0&0&\color{orangered}{ -8 }&3\\& & -16& 56& -224& \color{orangered}{896} & \\ \hline &4&-14&56&-224&\color{orangered}{888}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 888 } = \color{blue}{ -3552 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&4&2&0&0&-8&3\\& & -16& 56& -224& 896& \color{blue}{-3552} \\ \hline &4&-14&56&-224&\color{blue}{888}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -3552 \right) } = \color{orangered}{ -3549 } $
$$ \begin{array}{c|rrrrrr}-4&4&2&0&0&-8&\color{orangered}{ 3 }\\& & -16& 56& -224& 896& \color{orangered}{-3552} \\ \hline &\color{blue}{4}&\color{blue}{-14}&\color{blue}{56}&\color{blue}{-224}&\color{blue}{888}&\color{orangered}{-3549} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}-14x^{3}+56x^{2}-224x+888 } $ with a remainder of $ \color{red}{ -3549 } $.