Question

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

Answer 1

`(2x^3+6x^2-2x-5) div (x+3) = 2x^2-2 1/(x+3)`

Explanation:

`(color(blue)(2)x^3color(blue)(+6)x^2color(blue)(-2)xcolor(blue)(+3)) div (xcolor(red)(+)color(green)(3))`

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 `(-3)` 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 (`color(cyan)(1)`) is the remainder.
The preceding entries are the coefficients of the quotient expression: (`color(orange)(2)x^2color(orange)(+0)xcolor(orange)(-2)`)