Divide using synthetic division $$ ( 3x^{3}+16x^{2}+10x+25 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&3&16&10&25\\& & -15& -5& \color{black}{-25} \\ \hline &\color{blue}{3}&\color{blue}{1}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+16x^{2}+10x+25 }{ x+5 } = \color{blue}{3x^{2}+x+5} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&16&10&25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 3 }&16&10&25\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&16&10&25\\& & \color{blue}{-15} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-5&3&\color{orangered}{ 16 }&10&25\\& & \color{orangered}{-15} & & \\ \hline &3&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&16&10&25\\& & -15& \color{blue}{-5} & \\ \hline &3&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-5&3&16&\color{orangered}{ 10 }&25\\& & -15& \color{orangered}{-5} & \\ \hline &3&1&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&16&10&25\\& & -15& -5& \color{blue}{-25} \\ \hline &3&1&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&3&16&10&\color{orangered}{ 25 }\\& & -15& -5& \color{orangered}{-25} \\ \hline &\color{blue}{3}&\color{blue}{1}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+x+5 } $ with a remainder of $ \color{red}{ 0 } $.