Polynomial roots or discriminant

In summary, the conversation discusses the possibility of finding the roots of a given polynomial and the application of the Abel-Ruffini theorem in determining them. The speaker also inquires about obtaining the discriminant of the polynomial.
  • #1
nergal
2
0
Hi there,

I was wondering if it is possible to find the roots of the following polynomial
[tex]
P(x)=x^n+a x^m+b
[/tex]
or at least can I get the discriminant of it, which is the determinant of the Sylvester matrix associated to P(x) and P'(x).

Thanks
 
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  • #2
nergal said:
Hi there,

I was wondering if it is possible to find the roots of the following polynomial
[tex]
P(x)=x^n+a x^m+b
[/tex]

The Abel-Ruffini theorem says that it's impossible to find the roots of ##x^5 - x +1## using only rational numbers and radicals and combinations of those. So that's a counterexample to being able to find all the roots.
 
  • #3
Yes that's true, then do you have any idea how to get the discriminant ?
 

1. What is a polynomial?

A polynomial is a mathematical expression consisting of a sum of terms, each term being a constant multiplied by one or more variables raised to non-negative integer powers.

2. What are polynomial roots?

Polynomial roots are the values of the variables that make the polynomial equation equal to zero. In other words, they are the solutions to the polynomial equation.

3. What is the discriminant of a polynomial?

The discriminant of a polynomial is a value that is used to determine the number and nature of its roots. It is calculated using the coefficients of the polynomial and can be positive, negative, or zero.

4. How do you find the roots of a polynomial?

The roots of a polynomial can be found by factoring the polynomial or by using the quadratic formula. The quadratic formula is used for polynomials with degree two or higher and gives both real and complex roots.

5. What does the discriminant tell us about the roots of a polynomial?

The discriminant can tell us if the polynomial has two real roots, one real root, or two complex roots. If the discriminant is positive, the polynomial has two real roots. If the discriminant is zero, the polynomial has one real root. If the discriminant is negative, the polynomial has two complex roots.

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