Cubic with integer roots

For students of class 11-12 (age 16+)
User avatar
sowmitra
Posts: 155
Joined: Tue Mar 20, 2012 12:55 am
Location: Mirpur, Dhaka, Bangladesh

Cubic with integer roots

Unread post by sowmitra » Wed Dec 19, 2012 9:32 pm

Suppose that the polynomial $P(x)=x^3-ax^2+bx-c$ has $3$ positive integer roots and that $4a+2b+c=1741$. Determine the value of $a$.
"Rhythm is mathematics of the sub-conscious."
Some-Angle Related Problems;

User avatar
Phlembac Adib Hasan
Posts: 1016
Joined: Tue Nov 22, 2011 7:49 pm
Location: 127.0.0.1
Contact:

Re: Cubic with integer roots

Unread post by Phlembac Adib Hasan » Thu Dec 20, 2012 12:54 pm

Hint:
$1741$ is a prime. If it doesn't help then try to factorize LHS by adding something. I myself was able to find all the roots.
Welcome to BdMO Online Forum. Check out Forum Guides & Rules

User avatar
Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm
Location: Khulna,Bangladesh.

Re: Cubic with integer roots

Unread post by Tahmid Hasan » Sat Dec 22, 2012 6:30 pm

Let the roots of the equation be $p,q,r$.
So $p+q+r=a,pq+qr+rp=b,pqr=c$.
Since we are dealing with symmetric expressions, WLOG $p \le q \le r$
$4a+2b+c=(p+2)(q+2)(r+2)-8$
So $(p+2)(q+2)(r+2)=1749=3*11*53$
Hence $(p,q,r)=(1,9,51)$ so $a=61$
বড় ভালবাসি তোমায়,মা

Post Reply