Question 58
Consider the second degree polynomial $x^2 + a x + b$ 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 $x^4 + a x^3 + b x^2 + c x + d$ can be written as the product of four 1st degree polynomials if and only if $P(a, b, c, d) \ge 0$. (All the coefficients are real numbers.)