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From the root of cubic polynomial (Posted on 2023-09-16) Difficulty: 4 of 5
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).

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: Solution | Comment 2 of 4 |
(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
  Posted by Steve Herman on 2023-09-16 21:02:10

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