What is the sum of non-real roots of the polynomial equation?

Question

What is the sum of non-real roots of the polynomial equation $X^{3}+3X^{2}+3X+3=0$ ?

What have u tried so far? – user107676 Jan 01 '14 at 10:59 — user107676, Jan 01 '14 at 10:59

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

1

HINT:

$$x^3+3x^2+3x+3=(x+1)^3+2$$

In case you don't know about the roots of $y^3=-2,$ you can use this

edited Apr 13 '17 at 12:21

Community

answered Jan 01 '14 at 11:00

lab bhattacharjee

score 1 · Accepted Answer · answered Jan 01 '14 at 11:04

Let $f(x)$ be the left hand side of your equation.

Since $$(x+1)^3=-2,$$ we know one real solution is $$x=-1-\sqrt[3]{2}.$$

We know the other solutions are non-real solutions(Why?).

Now by Vieta's formulas, $$\alpha+\beta+\gamma=-3$$ where $\alpha,\beta,\gamma$ are the solutions of $f(x)=0.$

Then?

2 Answers2