Divide using synthetic division $$ ( -32x^{2}+288 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-32&0&288\\& & -96& \color{black}{-288} \\ \hline &\color{blue}{-32}&\color{blue}{-96}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -32x^{2}+288 }{ x-3 } = \color{blue}{-32x-96} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&-32&0&288\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -32 }&0&288\\& & & \\ \hline &\color{orangered}{-32}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ -96 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-32&0&288\\& & \color{blue}{-96} & \\ \hline &\color{blue}{-32}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -96 \right) } = \color{orangered}{ -96 } $
$$ \begin{array}{c|rrr}3&-32&\color{orangered}{ 0 }&288\\& & \color{orangered}{-96} & \\ \hline &-32&\color{orangered}{-96}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -96 \right) } = \color{blue}{ -288 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-32&0&288\\& & -96& \color{blue}{-288} \\ \hline &-32&\color{blue}{-96}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 288 } + \color{orangered}{ \left( -288 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}3&-32&0&\color{orangered}{ 288 }\\& & -96& \color{orangered}{-288} \\ \hline &\color{blue}{-32}&\color{blue}{-96}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -32x-96 } $ with a remainder of $ \color{red}{ 0 } $.