Divide using synthetic division $$ ( 3x^{4}-13x^{3}-14x^{2}+23x-15 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&3&-13&-14&23&-15\\& & 15& 10& -20& \color{black}{15} \\ \hline &\color{blue}{3}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-13x^{3}-14x^{2}+23x-15 }{ x-5 } = \color{blue}{3x^{3}+2x^{2}-4x+3} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-13&-14&23&-15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 3 }&-13&-14&23&-15\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-13&-14&23&-15\\& & \color{blue}{15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 15 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}5&3&\color{orangered}{ -13 }&-14&23&-15\\& & \color{orangered}{15} & & & \\ \hline &3&\color{orangered}{2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 2 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-13&-14&23&-15\\& & 15& \color{blue}{10} & & \\ \hline &3&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 10 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}5&3&-13&\color{orangered}{ -14 }&23&-15\\& & 15& \color{orangered}{10} & & \\ \hline &3&2&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-13&-14&23&-15\\& & 15& 10& \color{blue}{-20} & \\ \hline &3&2&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}5&3&-13&-14&\color{orangered}{ 23 }&-15\\& & 15& 10& \color{orangered}{-20} & \\ \hline &3&2&-4&\color{orangered}{3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&-13&-14&23&-15\\& & 15& 10& -20& \color{blue}{15} \\ \hline &3&2&-4&\color{blue}{3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 15 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&3&-13&-14&23&\color{orangered}{ -15 }\\& & 15& 10& -20& \color{orangered}{15} \\ \hline &\color{blue}{3}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+2x^{2}-4x+3 } $ with a remainder of $ \color{red}{ 0 } $.