# How do you divide 2x^3 + 6x^2 - 2x - 5 by x + 3 using synthetic division?

Jan 11, 2016

$\left(2 {x}^{3} + 6 {x}^{2} - 2 x - 5\right) \div \left(x + 3\right) = 2 {x}^{2} - 2 \frac{1}{x + 3}$

#### Explanation:

$\left(\textcolor{b l u e}{2} {x}^{3} \textcolor{b l u e}{+ 6} {x}^{2} \textcolor{b l u e}{- 2} x \textcolor{b l u e}{+ 3}\right) \div \left(x \textcolor{red}{+} \textcolor{g r e e n}{3}\right)$

would be set-up for synthetic division as:

{: (,"|",color(white)("X")color(blue)(2),color(blue)(+6),color(blue)(-2),color(blue)(-5)), (,"|",,,,), (bar(xx (color(red)(-)color(green)(3))),"|",bar(color(white)("X")color(orange)(2)color(white)("X")),bar(color(white)("XX")),bar(color(white)("XX")),bar(color(white)("XX"))) :}

For each column
- write the product of $\left(- 3\right)$ and the bottom entry of the previous column on the second line of the next column.
- add the entries in the next column to get the bottom line value for the next column.

{: (,"|",color(white)("X")color(blue)(2),color(blue)(+6),color(blue)(-2),color(blue)(-5)), (,"|",,-6,color(white)("X")0,color(white)("X")6), (bar(xx (color(red)(-)color(green)(3))),"|",bar(color(white)("X")color(orange)(2)color(white)("X")),bar(color(white)("X")color(orange)(0)),bar(color(orange)(-2)),bar(color(white)("X")color(cyan)(1))) :}

The last entry in the bottom line ($\textcolor{c y a n}{1}$) is the remainder.
The preceding entries are the coefficients of the quotient expression: ($\textcolor{\mathmr{and} a n \ge}{2} {x}^{2} \textcolor{\mathmr{and} a n \ge}{+ 0} x \textcolor{\mathmr{and} a n \ge}{- 2}$)