help, with polynomial question!! (1 Viewer)

[Jason Xie]

The euqation x^4 - x^3 + 2x^2 -2x +1 = 0 has roots, a, b, c, d
show that none of a, b, c, a, d is an integer.

Thank you so much, guys!!^-^

 

[Alkenes]

Find abcd and prove they are not integers

 

[Jason Xie]

any ways to do it without finding the roots??? thank you!!

 

[Alkenes]

dnt think so, havent touched ploy since last term lol

any ways to do it without finding the roots??? thank you!!

 

[Carrotsticks]

Find abcd and prove they are not integers

A good thought I guess, but quite 'brute force-ish'. And how would you exactly go about finding all 4 solutions =p

Since '1' is the constant term, this means that at least one of the root(s) must be a factor of 1.

So we test the following:

P(1) = ...

P(-1) = ...

We then observe that neither of them is equal to 0 ie: none of the zeroes are integers.

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Generally when trying to determine is a polynomial P(x) has integral zeroes or not, we substitute the factors of the CONSTANT term.

So say the constant term is 6, we would then test:

P(1)
P(-1)
P(6)
P(-6)
P(2)
P(-2)
P(3)
P(-3)

If none of them are equal to 0, then it has no integral zeroes.

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Now suppose the polynomial is NOT monic ie: has a leading coefficient.

We then test the factors of Constant term/Leading coefficient.

So suppose we have the polynomial P(x) =2x^3 - blah blah + 6, then we would test factors of 3 because 6/2 = 3.

So we can deduce from this that suppose the constant term is PRIME and the polynomial is NOT monic, then it has no integral zeroes.

 

[Jason Xie]

thanks, Carrotsticks,^-^

 

[deterministic]

This is what Carrotsticks is referring to: http://en.wikipedia.org/wiki/Rational_root_theorem