Question

How do you factor `2x^3 - 3x^2 - 5x`?

Answer 1

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

Explanation:

First, note that each term has a factor of `x`, and so we have
`2x^3 - 3x^2 - 5x = x(2x^2 - 3x - 5)`

Now, we can use the quadratic formula to find the remaining factors, but first let's see if there are easy integer solutions by looking for `a,b,c,d` where `2x^2 - 3x - 5 = (ax+b)(cx+d)`

We know that `ac = 2` and so we can look at `(2x+b)(x+d)`

We also know `bd = -5` and so our possible choices are `(2x+1)(x-5)` and `(2x-5)(x+1)`

Multiplying these out shows that `2x^2 - 3x - 5 = (2x-5)(x+1)`

So our final result is
`2x^3 - 3x^2 - 5x = x(2x-5)(x+1)`