Divide using synthetic division $$ ( 2x^{3}-10x^{2}+14x-24 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&2&-10&14&-24\\& & 8& -8& \color{black}{24} \\ \hline &\color{blue}{2}&\color{blue}{-2}&\color{blue}{6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-10x^{2}+14x-24 }{ x-4 } = \color{blue}{2x^{2}-2x+6} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{4}&2&-10&14&-24\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 2 }&-10&14&-24\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&2&-10&14&-24\\& & \color{blue}{8} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 8 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}4&2&\color{orangered}{ -10 }&14&-24\\& & \color{orangered}{8} & & \\ \hline &2&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&2&-10&14&-24\\& & 8& \color{blue}{-8} & \\ \hline &2&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}4&2&-10&\color{orangered}{ 14 }&-24\\& & 8& \color{orangered}{-8} & \\ \hline &2&-2&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&2&-10&14&-24\\& & 8& -8& \color{blue}{24} \\ \hline &2&-2&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}4&2&-10&14&\color{orangered}{ -24 }\\& & 8& -8& \color{orangered}{24} \\ \hline &\color{blue}{2}&\color{blue}{-2}&\color{blue}{6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-2x+6 } $ with a remainder of $ \color{red}{ 0 } $.