Let P(x) be a polynomial with degree 3, consider the polynomial
Q(x)=(x3-2x+1-P(x))(2x3-5x2+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
re: Solution
