Divide using synthetic division $$ ( 3x^{4}-18x^{3}+15x^{2}-x+5 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&3&-18&15&-1&5\\& & 15& -15& 0& \color{black}{-5} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-18x^{3}+15x^{2}-x+5 }{ x-5 } = \color{blue}{3x^{3}-3x^{2}-1} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-18&15&-1&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 3 }&-18&15&-1&5\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-18&15&-1&5\\& & \color{blue}{15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 15 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}5&3&\color{orangered}{ -18 }&15&-1&5\\& & \color{orangered}{15} & & & \\ \hline &3&\color{orangered}{-3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-18&15&-1&5\\& & 15& \color{blue}{-15} & & \\ \hline &3&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&3&-18&\color{orangered}{ 15 }&-1&5\\& & 15& \color{orangered}{-15} & & \\ \hline &3&-3&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-18&15&-1&5\\& & 15& -15& \color{blue}{0} & \\ \hline &3&-3&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 0 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}5&3&-18&15&\color{orangered}{ -1 }&5\\& & 15& -15& \color{orangered}{0} & \\ \hline &3&-3&0&\color{orangered}{-1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-18&15&-1&5\\& & 15& -15& 0& \color{blue}{-5} \\ \hline &3&-3&0&\color{blue}{-1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&3&-18&15&-1&\color{orangered}{ 5 }\\& & 15& -15& 0& \color{orangered}{-5} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-3x^{2}-1 } $ with a remainder of $ \color{red}{ 0 } $.