33Ok...
Can pi be teh root of a polynomial with integer coefficients??
Why why not?
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9 Answers
33No Pi can't be a root of polynomial with integers coefficients (or i think with rational coefficients)as....
(mere paas sabd nahi hai bolne ke liye....)
as a0pin can't be equal to other powers of pi multiplied by an integer or even rational nos.
consider a simple case:
a0xn+apxp=0; n>p.
and let pi be its root.
then anpin+appip=0
anpin-p+ap=0
pin-p=-ap/an which is not true as p is not equal to n
Don't know the general proof...
62hmm.. ur understanding is ok...
basically this will never be asked in JEE.. but it is very useful for basics and understanding of mathematics...
U may want to read Transcendental number.. it is a good thing to know..
62btw u will be surprised to know that it is not known whether
π + e and π − e are rationals or irrationals!
1pi cant be a root coz....
pi is an irrational no.
and any quad eqn with rational coeffs have all rational roots.... haina?
62no.. root 2 !
is the root of x2=2
But this is not relevant to JEE .. or any entrance exam.. (more for ISI or CMI or Olympiad)
So if u dont understand.. dont worry :)
1no no plz make me understand....... its interesting.... i wanna know.... plz
62basically there is a class of Transcendental Numbers...
These numbers are not the solution of any polynomial...
This whole set is very large... Many more than the rational number
(I mean there are more numbers (real) which are not the roots of a polynomial... than those that are roots) (ofcourse both these are infinite! but there is a way to compare infinites.. dont go into that)
I guess knowing this much should suffice :)
btw repeating what i said in an earlier post.....
we(even mathematicians) dont know if π + e and π − e are rationals or irrationals!
infact about any combination a.Ï€ +b.e (a, b are integers)
62if u like this question, u may like
http://targetiit.com/iit_jee_forum/posts/prove_divisibilty_by_9_149.html
as well