Cubic with integer roots

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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 $a$.
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Phlembac Adib Hasan
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Re: Cubic with integer roots

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

$1741$ 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.
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Tahmid Hasan
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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 $p \le q \le r$
So $(p+2)(q+2)(r+2)=1749=3*11*53$
Hence $(p,q,r)=(1,9,51)$ so $a=61$
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