Let \(P(x)\) be a polynomial whose coefficients are either \(1\) or \(-1\). For example \(P(x)\) can be \(x^2+x-1\). Given that the roots of \(P\) are all real, prove that \(\deg P\le 3\). FYI \(\deg P\) denotes the degree of \(P(x)\).
PC: Mursalin Habib. (memberlist.php?mode=viewprofile&u=2242)
Coefficient Restriction
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: Coefficient Restriction
Is the constant term considered as a coefficient?
Re: Coefficient Restriction
Yes of course.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Phlembac Adib Hasan
- Posts:1016
- Joined:Tue Nov 22, 2011 7:49 pm
- Location:127.0.0.1
- Contact:
Re: Coefficient Restriction
Misinterpreted. The roots should be distinct. Otherwise counter-examples like $P(x)=x^9-x^8$ can be constructed easily.Nirjhor wrote:Let \(P(x)\) be a polynomial whose coefficients are either \(1\) or \(-1\). For example \(P(x)\) can be \(x^2+x-1\). Given that the roots of \(P\) are all real, prove that \(\deg P\le 3\). FYI \(\deg P\) denotes the degree of \(P(x)\).
PC: Mursalin Habib. (memberlist.php?mode=viewprofile&u=2242)
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Re: Coefficient Restriction
That's not a valid construction. You are considering the coefficients of \(x^i\) for \(i\in\{0,1,...,7\}\) to be \(0\), which is not allowed.Phlembac Adib Hasan wrote:Misinterpreted. The roots should be distinct. Otherwise counter-examples like $P(x)=x^9-x^8$ can be constructed easily.Nirjhor wrote:Let \(P(x)\) be a polynomial whose coefficients are either \(1\) or \(-1\). For example \(P(x)\) can be \(x^2+x-1\). Given that the roots of \(P\) are all real, prove that \(\deg P\le 3\). FYI \(\deg P\) denotes the degree of \(P(x)\).
PC: Mursalin Habib. (memberlist.php?mode=viewprofile&u=2242)
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
Re: Coefficient Restriction
Since this is going unsolved for too long.
Hint 1 Hint 2 My Solution
Hint 1
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.