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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 :)

  • 0
    If a polynomial $P$ has roots $a, b$ and a< b, then $P$ will attain a minimum or maximum in $[a,b]$.2012-01-06

1 Answers 1

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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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    Factor $P'$ as $x(x^{n-2}+2a)$. Since $n-2$ is odd, $P'$ has exactly 2 real roots.2012-01-07