Question

How do you factor 2x³−3x²−4x-4?

Answer 1

`2(x - 2.57716869)(x^2 + 1.0771687 x + 0.776046)`

Explanation:

There is no easy factorization here.
There is a method to solve a cubic equation in general by hand (and calculator) on paper.
This method is based on the substitution of Vieta method.
Dividing by the first coefficient yields :
`x^3 - (3/2) x^2 - 2 x - 2 = 0`
Substituting x=y+p in `x^3+ax^2+bx+c=0` yields :
`y^3 + (3p+a) y^2 + (3p^2+2ap+b) y + p^3+ap^2+bp+c = 0`
if we take 3p+a=0 or p=-a/3, the first coefficient becomes zero, and we get :
`y^3 - (11/4) y - (13/4) = 0`
(with p = 1/2)
Substituting y=qz in `y^3 + b y + c = 0`, yields :
`z^3 + b z / q^2 + c / q^3 = 0`
if we take q = sqrt(|b|/3), the coefficient of z becomes 3 or -3, and we get :
(here q = 0.95742711)
`z^3 - 3 z - 3.70310650 = 0`
Substituting z = t + 1/t, yields :
`t^3 + 1/t^3 - 3.70310650 = 0`
Substituting `u = t^3`, yields the quadratic equation :
`u^2 - 3.70310650 u + 1 = 0`
A root of this quadratic equation is u=3.40983738.
Substituting the variables back, yields :
t = cuberoot(u) = 1.50514344.
z = 2.16953194.
y = 2.07716869.
`x = 2.57716869`.
The other roots can be found by dividing and solving the remaining quadratic equation.
The other roots are complex : `-0.53858435 pm 0.69711716 i`.