Cubic with integer roots

[sowmitra]

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 $a$.

[Phlembac Adib Hasan]

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.

[Tahmid Hasan]

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 $a=61$