Find the equation of cubic polynomial

In summary, the equation of the cubic polynomial can be found by considering the roots α, β, and γ and using Vieta's formulas or Newton's identities. The equation is in the form x3 - (α+β+γ)x2 + (αβ + αγ + βγ)x - αβγ = 0. However, there is no easy way to solve a cubic polynomial unless the roots are known to be integers.
  • #1
songoku
2,266
319

Homework Statement


If α, β and γ are roots of cubic polynomial and:
αβγ = 6
α222=20
α333=121

Find the equation of cubic polynomial


Homework Equations


vieta


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
 
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  • #2
songoku said:
If α, β and γ are roots of cubic polynomial …

oh come on songoku! …

(x-α)(x-β)(x-γ) ? :wink:
 
  • #3
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
songoku said:
But I don't know how to find α+β+γ and αβ + αγ + βγ. Thanks

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
Hi songoku! :smile:

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
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:
  • #7
hi songoku! :smile:

your method looks fine

there's no easy way to solve a cubic polynomial (unless you know the roots are integers) :redface:
 
  • #8
tiny-tim said:
hi songoku! :smile:

your method looks fine

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

hi tiny-tim :smile:

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

thanks a lot for the help :smile:
 

What is a cubic polynomial?

A cubic polynomial is a type of algebraic expression that involves the variable raised to the third power. It can be written in the form of ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

How do I find the equation of a cubic polynomial?

To find the equation of a cubic polynomial, you need to know four points on the graph of the polynomial. Then, you can use those points to create a system of equations and solve for the coefficients a, b, c, and d.

What is the degree of a cubic polynomial?

The degree of a cubic polynomial is 3, since it is raised to the third power. This means that the highest power of the variable in the polynomial is 3.

What are the key features of a cubic polynomial?

The key features of a cubic polynomial include the leading coefficient (a), the constant term (d), the x-intercepts (where the polynomial crosses the x-axis), and the turning points (where the polynomial changes direction).

How do I graph a cubic polynomial?

To graph a cubic polynomial, you can use a graphing calculator or manually plot points using the equation of the polynomial. You can also identify the key features of the polynomial and use them to sketch a rough graph.

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