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I have a homework question which is:

True or False: if $n$ is not even then $P(x)=x^n+ax^2+b$ has at most 3 roots

I know that the version of $n$ being even is true via some recursion and solving a squared function you will get 2 roots

But I can't seem to see if this one is true or false.

Can some one please help me?

Thank a-lot :)

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If a polynomial $P$ has roots $a, b$ and $a< b$, then $P$ will attain a minimum or maximum in $[a,b]$. –  user20266 Jan 6 '12 at 15:40
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1 Answer 1

up vote 11 down vote accepted

If $n=1$, the statement is obviously true. Consider odd number $n\geq 3$. Assume that $P(x)$ have $4$ roots or more, then by Rolle's theorem $$ P'(x)=nx^{n-1}+2ax $$ have at least $3$ roots. Then $$ P''(x)=n(n-1)x^{n-2}+2a $$ have at least $2$ roots. The last result is impossible since $n-2$ is odd.

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Ah Great :) Thank you –  Jason Jan 6 '12 at 15:51
    
Factor $P'$ as $x(x^{n-2}+2a)$. Since $n-2$ is odd, $P'$ has exactly 2 real roots. –  Chris Leary Jan 7 '12 at 4:36
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