Let P(x) be a polynomial with degree 3, consider the polynomial

Q(x)=(x^{3}-2x+1-P(x))(2x^{3}-5x^{2}+4-P(x)).

Assume that Q(x)≤0, ∀x and P(0)=3. Calculate Q(-1).

(In reply to

Solution by Larry)

Nicely done, Larry. I believe that you have the only solution.

If

f1(x) = x^3-2x+1 and

f2(x) = 2x^3-5x^2+4,

then the functions intersect when f2(x)-f1(x) = x^3-5x^2+2x+3 = 0. That cubic has three distinct solutions, not 2, at -0.57577, 1.18728 and 4.38849. So three points of intersection, not two. and P(0) = 3 provides a 4th point that P(x) must honor, so P(x) is uniquely determined

*Edited on ***September 16, 2023, 9:15 pm**