finding the product of $z_1z_2z_3$ if $(z+m)^3=-27i$

Question

in complex numbers, if:

$(z+m)^3=-27i$

Need to find the term $z_1z_2z_3$ and the term $z_1+z_2+z_3$ by $m$.

While $z_1,z_2,z_3$ are the roots of the equation.

Tried to simplify but got messy. i thought about vieta formulas but we didn't learn that for cubic equations maybe it can be done without that?

Thanks

Vieta's Formulas can be applied. – Ahaan S. Rungta Nov 20 '13 at 17:40 — Ahaan S. Rungta, Nov 20 '13 at 17:40

score 1 · Answer 1 · answered Nov 20 '13 at 17:41

Your cubic equation is

$$z^3 +3mz^2 +3m^2z + m^3 = -27i$$

or

$$z^3 +3mz^2 +3m^2z + m^3 +27i = 0$$

The sum of the roots is then $-3m$ and the product of the 3 roots is $-m^3-27i$

Even if you are unsure of the Vieta formulae, you can see this quite easily by comparing with the expansion of $$(z-z_1)(z-z_2)(z-z_3)$$

1 Answers1