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