MathDB

Consider the second degree polynomial

x^2+ax+b

with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant,

a^2-4b

be greater than or equal to zero. Note that the discriminant is also a polynomial with variables

a

and

b

. Prove that the same story is not true for polynomials of degree

4

: Prove that there does not exist a

4

variable polynomial

P(a,b,c,d)

such that:
The fourth degree polynomial

x^4+ax^3+bx^2+cx+d

can be written as the product of four

1

st degree polynomials if and only if

P(a,b,c,d)\ge 0

. (All the coefficients are real numbers.)
Proposed by Sahand Seifnashri

Problem Statement