# Find the equation of cubic polynomial

1. May 20, 2012

### songoku

1. The problem statement, all variables and given/known data
If α, β and γ are roots of cubic polynomial and:
αβγ = 6
α222=20
α333=121

Find the equation of cubic polynomial

2. Relevant equations
vieta

3. The attempt at a solution
The equation is in the form:
x3 - (α+β+γ)x2 + (αβ + αγ + βγ) x - αβγ = 0

But I don't know how to find α+β+γ and αβ + αγ + βγ. Thanks

2. May 20, 2012

### tiny-tim

oh come on songoku! …

(x-α)(x-β)(x-γ) ?

3. May 20, 2012

### Joffan

Three equations, three unknowns... a bit of devil to solve easily though.

For the first equation in αβγ space, I'm seeing four hyperbolic surfaces in +++ and three +-- octants
The second equation a sphere of course - this tells me the magnitudes of αβγ are all less than √20 < 4.5
The third equation therefore limits me to the +++ case (121 is nearly 5^3)...

4. May 20, 2012

### rock.freak667

You are given what α222 is. So what happens if you consider what (α+β+γ)2 is? (expand it out and see what terms you have)

5. May 20, 2012

### I like Serena

Hi songoku!

Related to Vieta's formulas are Newton's identities.
They work out the same as rock.freak667's suggestion.

From these you can get a relation between your unknown coefficients and the equations that you are given.
You won't find nice round numbers though.

6. May 20, 2012

### songoku

There are some posts above that I don't actually understand but let me try

let : α+β+γ = p ; αβ + αγ + βγ = q

(α+β+γ)2 = α2 + β2 + γ2 + 2 (αβ + αγ + βγ)
p2 = 20 + 2q
q = 1/2 (p2 - 20)

(α+β+γ)3 = α3 + β3 + γ3 + 3(α+β+γ)(αβ + αγ + βγ) - 3 αβγ
p3 = 121 + 3pq - 18

subs. q to second equation results in cubic equation in terms of p, then by using calculator I got p = -6.7

Am I correct? How to find p manually?

Thanks

Last edited: May 21, 2012
7. May 21, 2012

### tiny-tim

hi songoku!

there's no easy way to solve a cubic polynomial (unless you know the roots are integers)

8. May 21, 2012

### songoku

hi tiny-tim

*sigh...* hope that the root will be integer when exam comes....

thanks a lot for the help