Question

How do you factor `3x ^ { 5} + 9x ^ { 4} - 6x` completely?

Answer 1

`3x(x+3.0692)(x-0.8069)(x^2+0.7377x+0.8076)`

Explanation:

= `3x(x^4 + 3x^3 - 2)`
There is a method to solve a quartic equation in general by hand (and calculator) on paper.
This method is called 'the method of Descartes'.
Substituting x=y+p in `x^4+ax^3+bx^2+cx+d` yields :
`y^4 + (4p+a) y^3 + (6p^2+3ap+b) y^2 + (4p^3+3ap^2+2bp+c) y + p^4+ap^3+bp^2+cp+d`
if we take 4p+a=0 or p=-a/4, the coefficient of y³ becomes zero, and we get :
`y^4 - (27/8) y^2 + (27/8) y - (755/256)`
(with p = -3/4)
The method of Descartes writes `y^4+by^2+cy+d` as `(y^2+ky+m)(y^2-ky+n)`
elimination of m and n results in the cubic equation
`x^3+2bx^2+(b^2-4d)x-c^2=0`,
with x = k², so once we have found positive x, we know k,m,n and the solutions of the quartic equation as solutions of two quadratic equations.
In our example this cubic equation is :
`x^3 - (27/4) x^2 + (371/16) x - (729/64) = 0`
There is a method to solve a cubic equation in general by hand (and simple calculator) on paper.
Dividing by the first coefficient yields :
`x^3 - (27/4) x^2 + (371/16) x - (729/64)`
Substituting x=y+p in `x^3+ax^2+bx+c` yields :
`y^3 + (3p+a) y^2 + (3p^2+2ap+b) y + p^3+ap^2+bp+c`
if we take 3p+a=0 or p=-a/3, the first coefficient becomes zero, and we get :
`y^3 + 8 y + 18 = 0`
(with p = 9/4)
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 = 1.63299316)
`z^3 + 3 z + 4.13351394 = 0`
Substituting z = t - 1/t, yields :
`t^3 - 1/t^3 + 4.13351394 = 0`
Substituting u = t³, yields the quadratic equation :
`u^2 + 4.13351394 u - 1 = 0`
A root of this quadratic equation is u=0.22921437.
Substituting the variables back, yields :
t = cuberoot(u) = 0.61199416.
z = -1.02200836.
y = -1.66893267.
x = 0.58106733.
The other roots of the cubic can be found by dividing and solving the remaining quadratic equation.
Filling in this value for k² yields as roots :
`-3.06917737 and 0.80689964`.
`-0.36886114 +- 0.81946746 i`
So the factorization is
`3x(x+3.0692)(x-0.8069)(x^2+0.7377x+0.8076)`.

Note that here there is a shortcut possible if we use y=1/x as substitution then we have a quartic with first coefficient already zero and yields in a cubic resolvent with first coefficient also zero already. I did not use the shortcut to show you the general method of Descartes.