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