IIT JEE , AIEEE Forum - Mathematics , Algebra

hsbhatt

Let P(x) be a polynomial with integral coefficients. Prove that we can never have

P(a) = b and P(b) = c and P(c) = a where a,b and c are integers.

hsbhatt

Anybody have a look at this yet?

konichiwa2x

Are you sure that's the entire question, sir?

How about P(x) has no constant term and a = b = c = 0?

hsbhatt

That's all there is to the question. And pls dont call me sir. bro is just fine by me.

konichiwa2x

well, i think the condition should be given that a, b, and c are distinct. Otherwise a=b=c=0 (considering a polynomial with no constants) satisifes. So contradicts.

Let

be a polynomial with integer coefficients and

and

be two integers. Then

is divisible by

----(1)

Proof:

now suppose we have

and

then by (1) we get,

and

so

and

, for some integers

and

So

Hence

Thus

and

That gives us

now by (1),

but

Thus

, which is impossible....answer follows.

hsbhatt

Great stuff!

And the forum must thank you for introducing this concept that P(a) - P(b) is div by (a-b). It is very useful in many problems but I encountered this funda only recently.

Good going. And thanx again for pointing out my carelessness in conditions.

konichiwa2x

thanks!

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