Iranian Mathematical Olympiad

Question

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) 0. (All the coefficients are real numbers.)

Accepted Answer

If we put a = c = 0, polynomial x^4 + b x^2 + d can be written as product of four linear terms if and only if quadratic polynomial y^2 + b y + d has two nonnegative roots. Therefore P(0, b, 0, d) 0 if and only if b 0, d 0 and b^2 - 4d 0. For a fixed b 0 let Q_b(d) = P(0, b, 0, d). Now, Q_b(d) 0 if and only if 0 d b^24 and hence by continuity of Q_b, Q_b(b^2/4) must be zero. This implies that for all b 0, one variable polynomial P(0, b, 0, b^24) = 0, and hence this polynomial is always zero. This means that polynomial x^4 + b x^2 + b^24 has four real roots for all values of b. Contradiction!