Let P(x) be a real quadratic trinomial, so that for all x∈R the inequality P(x³+x)≥P(x²+1) holds. Find the sum of the roots of P(x).

(In reply to Solution by Jer)

So I assume for your purposes p=x, and q= the 5^th order polynomial you derived via expanding the inequality and collecting the terms on the LHS.<o:p></o:p>

However, I’m not sure your deductions that follow are correct?<o:p></o:p>

Doesn’t pq>=0 imply the following:<o:p></o:p>

If p>0, then q>=0, not q>0<o:p></o:p>

If p<0, then q<=0, not q<0<o:p></o:p>

If p=0, then q=b<o:p></o:p>

If q=0, p is limited to any root of the 5^th order polynomial?<o:p></o:p>

or - my algebra is rusty - which could be true too!

Posted by Kenny M on 2021-04-02 16:11:44