Divide using synthetic division $$ ( 4x^{4}+2x^{3}-3x^{2}+1 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&4&2&-3&0&1\\& & 16& 72& 276& \color{black}{1104} \\ \hline &\color{blue}{4}&\color{blue}{18}&\color{blue}{69}&\color{blue}{276}&\color{orangered}{1105} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+2x^{3}-3x^{2}+1 }{ x-4 } = \color{blue}{4x^{3}+18x^{2}+69x+276} ~+~ \frac{ \color{red}{ 1105 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&2&-3&0&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 4 }&2&-3&0&1\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&2&-3&0&1\\& & \color{blue}{16} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 16 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrr}4&4&\color{orangered}{ 2 }&-3&0&1\\& & \color{orangered}{16} & & & \\ \hline &4&\color{orangered}{18}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 18 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&2&-3&0&1\\& & 16& \color{blue}{72} & & \\ \hline &4&\color{blue}{18}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 72 } = \color{orangered}{ 69 } $
$$ \begin{array}{c|rrrrr}4&4&2&\color{orangered}{ -3 }&0&1\\& & 16& \color{orangered}{72} & & \\ \hline &4&18&\color{orangered}{69}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 69 } = \color{blue}{ 276 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&2&-3&0&1\\& & 16& 72& \color{blue}{276} & \\ \hline &4&18&\color{blue}{69}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 276 } = \color{orangered}{ 276 } $
$$ \begin{array}{c|rrrrr}4&4&2&-3&\color{orangered}{ 0 }&1\\& & 16& 72& \color{orangered}{276} & \\ \hline &4&18&69&\color{orangered}{276}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 276 } = \color{blue}{ 1104 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&2&-3&0&1\\& & 16& 72& 276& \color{blue}{1104} \\ \hline &4&18&69&\color{blue}{276}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 1104 } = \color{orangered}{ 1105 } $
$$ \begin{array}{c|rrrrr}4&4&2&-3&0&\color{orangered}{ 1 }\\& & 16& 72& 276& \color{orangered}{1104} \\ \hline &\color{blue}{4}&\color{blue}{18}&\color{blue}{69}&\color{blue}{276}&\color{orangered}{1105} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+18x^{2}+69x+276 } $ with a remainder of $ \color{red}{ 1105 } $.