Problem
ALG-B1-M09-P013 Monic polynomial
#13
★★★☆☆ Level 3 of 5
Find all integers \(a\) for which the polynomial \(x^2+ax+12\) has two integer roots.
Let the roots be \(r,s\). Then \(rs=12\), \(r+s=-a\).
List integer factor pairs \(rs=12\): \((1,12),(2,6),(3,4)\) and permutations, plus negative pairs. The possible sums are \(13,8,7,-13,-8,-7\). Therefore \(a=-13,-8,-7,7,8,13\).
The parameter is found through Vieta and divisors of the constant term.