Divide using synthetic division $$ ( 2x^{4}-13x^{2}+3 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&2&0&-13&0&3\\& & -10& 50& -185& \color{black}{925} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{37}&\color{blue}{-185}&\color{orangered}{928} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-13x^{2}+3 }{ x+5 } = \color{blue}{2x^{3}-10x^{2}+37x-185} ~+~ \frac{ \color{red}{ 928 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&0&-13&0&3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 2 }&0&-13&0&3\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&0&-13&0&3\\& & \color{blue}{-10} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}-5&2&\color{orangered}{ 0 }&-13&0&3\\& & \color{orangered}{-10} & & & \\ \hline &2&\color{orangered}{-10}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&0&-13&0&3\\& & -10& \color{blue}{50} & & \\ \hline &2&\color{blue}{-10}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 50 } = \color{orangered}{ 37 } $
$$ \begin{array}{c|rrrrr}-5&2&0&\color{orangered}{ -13 }&0&3\\& & -10& \color{orangered}{50} & & \\ \hline &2&-10&\color{orangered}{37}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 37 } = \color{blue}{ -185 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&0&-13&0&3\\& & -10& 50& \color{blue}{-185} & \\ \hline &2&-10&\color{blue}{37}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -185 \right) } = \color{orangered}{ -185 } $
$$ \begin{array}{c|rrrrr}-5&2&0&-13&\color{orangered}{ 0 }&3\\& & -10& 50& \color{orangered}{-185} & \\ \hline &2&-10&37&\color{orangered}{-185}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -185 \right) } = \color{blue}{ 925 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&0&-13&0&3\\& & -10& 50& -185& \color{blue}{925} \\ \hline &2&-10&37&\color{blue}{-185}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 925 } = \color{orangered}{ 928 } $
$$ \begin{array}{c|rrrrr}-5&2&0&-13&0&\color{orangered}{ 3 }\\& & -10& 50& -185& \color{orangered}{925} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{37}&\color{blue}{-185}&\color{orangered}{928} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-10x^{2}+37x-185 } $ with a remainder of $ \color{red}{ 928 } $.