Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
if a polynomial P with integer coefficients has three distinct integer zeroes, then show that P(n) not equal to 1 for any integer n.
Dear Aritra,
The question is incorrect,
There are so many examples to prove it incorrect such as :
let P = (x-1)(x-2)(x-3)* (1/6 )
its zeroes are 1,2,3 which are integers and its value at 4(an integer) is 1.
Best Of luck
Cracking IIT just got more exciting,It s not just all about getting assistance from IITians, alongside Target Achievement and Rewards play an important role. ASKIITIANS has it all for you, wherein you get assistance only from IITians for your preparation and win by answering queries in the discussion forums. Reward points 5 + 15 for all those who upload their pic and download the ASKIITIANS Toolbar, just a simple to download the toolbar….
So start the brain storming…. become a leader with Elite Expert League ASKIITIANS
Thanks
Aman Bansal
Askiitian Expert
Hi Aritra,
The above example P(x) = 1/6*(x-1)(x-2)(x-3) does not have integer co-efficients.
Let P(x) = (x-a)(x-b)(x-c)*Q(x). where A,a,b,c are integers with a,b,c being distinct. Q(x) is any polynomial with integer co-efficients.
Now for any integer n, consider P(n).
P(n) = (n-a)(n-b)(n-c)Q(n).
In the above expression Q(n) is any integer.
Assume P(n) = 1 for some n.
Since all are integers, only possibility is |n-a| = |n-b| = |n-c| = |Q(n)| = 1.
Consider only |n-a|=|n-b|=|n-c|=1 ------------------ (1)
Expression (1) can never be true for distinct integers a,b,c as explained below.
|n-a|=|n-b|
Only possibility n-a = b-n, ie a+b=2n.
similarly b+c=2n.
Subtraction a-c=0 or a=c --------- (contradiction).
Hence P(n) can never be equal to 1 for any integer n.
Regards,
Ashwin (IIT Madras).
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Win Gift vouchers upto Rs 500/-
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !