Formula for the roots of a cubic equation

Question

I know that you can derive a quadratic formula from the given complex roots $\alpha$ and $\beta$ if you simply put them into the formula $x^2-(\alpha+\beta)x+\alpha\beta=0$. Is there an equivalent for cubic equations?

Answer 1

I think what you're asking is this: if you know the roots of a cubic polynomial, then can you get the expression for the polynomial itself? And the answer is you can: if a cubic has roots $\alpha, \beta, \gamma$, then the formula for the cubic is just $$(x - \alpha) (x - \beta) (x - \gamma)$$ which expands to $$ x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha \beta + \beta \gamma + \gamma \alpha) x - \alpha \beta \gamma.$$

The quadratic formula that you posted likewise, is just the expansion of $(x - \alpha)(x - \beta)$.