Solving a cubic polynomial equation.

polynomials roots transformation cubics

Solution 1:

I'll give you some hints.

1) Transform the equation to the form $x^3 + 3bx + c$. (The 3 just comes from a simple change of variable, but will come in handy). Your transformation method may come in handy here.

2) Now for the clever trick. Let $x = w - \frac{b}{w}$. This transforms the equation into $w^3 + c - \frac{b^3}{w^3} = 0$... which is a quadratic in $w^3$. So solve this

3) You now have a value for $w$. Can you now find a root of the equation?

Related

a question related to two competing patterns in coin tossing
Inequality.$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq 3$
What is an advatage of defining $\mathbb{C}$ as a set containing $\mathbb{R}$?
Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms
Space of bounded functions is reflexive if the domain is finite
How do I prove whether something is a Euclidean domain?
Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$
No closed form for the partial sum of ${n\choose k}$ for $k \le K$?
Transforming $2D$ outline into $3D$ plane
Finding a 2x2 Matrix raised to the power of 1000
fundamental group of $GL^{+}_n(\mathbb{R})$
Laplace transform of $ t^{1/2}$ and $ t^{-1/2}$