Divide using synthetic division $$ ( 2x^{6}-28x^{5}+53x^{4}-16x^{3}-60x^{2} ) \div ( 2x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-4&2&-28&53&-16&-60&0&0\\& & -8& 144& -788& 3216& -12624& \color{black}{50496} \\ \hline &\color{blue}{2}&\color{blue}{-36}&\color{blue}{197}&\color{blue}{-804}&\color{blue}{3156}&\color{blue}{-12624}&\color{orangered}{50496} \end{array} $$The solution is:
$$ \frac{ 2x^{6}-28x^{5}+53x^{4}-16x^{3}-60x^{2} }{ x+4 } = \color{blue}{2x^{5}-36x^{4}+197x^{3}-804x^{2}+3156x-12624} ~+~ \frac{ \color{red}{ 50496 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-4&\color{orangered}{ 2 }&-28&53&-16&-60&0&0\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & \color{blue}{-8} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrrrrr}-4&2&\color{orangered}{ -28 }&53&-16&-60&0&0\\& & \color{orangered}{-8} & & & & & \\ \hline &2&\color{orangered}{-36}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & -8& \color{blue}{144} & & & & \\ \hline &2&\color{blue}{-36}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 53 } + \color{orangered}{ 144 } = \color{orangered}{ 197 } $
$$ \begin{array}{c|rrrrrrr}-4&2&-28&\color{orangered}{ 53 }&-16&-60&0&0\\& & -8& \color{orangered}{144} & & & & \\ \hline &2&-36&\color{orangered}{197}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 197 } = \color{blue}{ -788 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & -8& 144& \color{blue}{-788} & & & \\ \hline &2&-36&\color{blue}{197}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -788 \right) } = \color{orangered}{ -804 } $
$$ \begin{array}{c|rrrrrrr}-4&2&-28&53&\color{orangered}{ -16 }&-60&0&0\\& & -8& 144& \color{orangered}{-788} & & & \\ \hline &2&-36&197&\color{orangered}{-804}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -804 \right) } = \color{blue}{ 3216 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & -8& 144& -788& \color{blue}{3216} & & \\ \hline &2&-36&197&\color{blue}{-804}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ 3216 } = \color{orangered}{ 3156 } $
$$ \begin{array}{c|rrrrrrr}-4&2&-28&53&-16&\color{orangered}{ -60 }&0&0\\& & -8& 144& -788& \color{orangered}{3216} & & \\ \hline &2&-36&197&-804&\color{orangered}{3156}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3156 } = \color{blue}{ -12624 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & -8& 144& -788& 3216& \color{blue}{-12624} & \\ \hline &2&-36&197&-804&\color{blue}{3156}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12624 \right) } = \color{orangered}{ -12624 } $
$$ \begin{array}{c|rrrrrrr}-4&2&-28&53&-16&-60&\color{orangered}{ 0 }&0\\& & -8& 144& -788& 3216& \color{orangered}{-12624} & \\ \hline &2&-36&197&-804&3156&\color{orangered}{-12624}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -12624 \right) } = \color{blue}{ 50496 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&-28&53&-16&-60&0&0\\& & -8& 144& -788& 3216& -12624& \color{blue}{50496} \\ \hline &2&-36&197&-804&3156&\color{blue}{-12624}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 50496 } = \color{orangered}{ 50496 } $
$$ \begin{array}{c|rrrrrrr}-4&2&-28&53&-16&-60&0&\color{orangered}{ 0 }\\& & -8& 144& -788& 3216& -12624& \color{orangered}{50496} \\ \hline &\color{blue}{2}&\color{blue}{-36}&\color{blue}{197}&\color{blue}{-804}&\color{blue}{3156}&\color{blue}{-12624}&\color{orangered}{50496} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}-36x^{4}+197x^{3}-804x^{2}+3156x-12624 } $ with a remainder of $ \color{red}{ 50496 } $.