Divide using synthetic division $$ ( 3x^{4}+14x^{3}-9x^{2}-22x-10 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&3&14&-9&-22&-10\\& & -15& 5& 20& \color{black}{10} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+14x^{3}-9x^{2}-22x-10 }{ x+5 } = \color{blue}{3x^{3}-x^{2}-4x-2} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&14&-9&-22&-10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 3 }&14&-9&-22&-10\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&14&-9&-22&-10\\& & \color{blue}{-15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-5&3&\color{orangered}{ 14 }&-9&-22&-10\\& & \color{orangered}{-15} & & & \\ \hline &3&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&14&-9&-22&-10\\& & -15& \color{blue}{5} & & \\ \hline &3&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 5 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-5&3&14&\color{orangered}{ -9 }&-22&-10\\& & -15& \color{orangered}{5} & & \\ \hline &3&-1&\color{orangered}{-4}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&14&-9&-22&-10\\& & -15& 5& \color{blue}{20} & \\ \hline &3&-1&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 20 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-5&3&14&-9&\color{orangered}{ -22 }&-10\\& & -15& 5& \color{orangered}{20} & \\ \hline &3&-1&-4&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&14&-9&-22&-10\\& & -15& 5& 20& \color{blue}{10} \\ \hline &3&-1&-4&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 10 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-5&3&14&-9&-22&\color{orangered}{ -10 }\\& & -15& 5& 20& \color{orangered}{10} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-x^{2}-4x-2 } $ with a remainder of $ \color{red}{ 0 } $.