Solving cubic equations with another method.

Question

I want to solve cubic equations with some pattern or formula. My maths teacher told me that if we want to solve cubic equations we have to find any one value for the equation which satisfy it and then divide the main equation by it. But I find that method little useless . Is there any other simplier or good method to solve it. You can explain it by an example.

Answer 1

For example.

Solve the following equation. $$x^3-3x+1=0.$$ Solution.

Let $x=2\cos\alpha$.

Thus, $$8\cos^3\alpha-6\cos\alpha+1=0$$ or $$\cos3\alpha=-\frac{1}{2}$$ or $$3\alpha=\pm120^{\circ}+360^{\circ}k,$$ where $k\in\mathbb Z$ or $$\alpha=\pm40^{\circ}+120^{\circ}k,$$ which gives the answer: $$\{2\cos40^{\circ},2\cos80^{\circ},-2\cos20^{\circ}\}.$$

Cubic equations are not so easy thing.

For example, one of roots of the equation $$x^3+x^2-10x-8=0$$ it's $$x_1=2\left(\cos\frac{2\pi}{31}+\cos\frac{4\pi}{31}+\cos\frac{8\pi}{31}+\cos\frac{16\pi}{31}+\cos\frac{30\pi}{31}\right).$$