Question

How do you factor `2x^3 + 7x^2 - 3x -18`?

Answer 1

`2x^3+7x^2-3x-18=(x+3)(x+2)(2x-3)`

The `2x^3` term quickly becomes dominant as `|x|` increases, so any solutions of `2x^3+7x^2-3x-18 = 0` will be quite small.

Integer solutions will be factors of 18, so try `x = -3, -2, -1, 0, 1, 2 or 3`. `x = -3` is a solution, so `(x + 3)` is a factor. `x = -2` is also a solution, so `(x+2)` is also a factor. If the remaining factor is of the form `(ax+b)` then looking at the coefficient of `x^3` (`2`) and the constant term (`18`) we must have `a=2` and `b=-3`.