2023 AMC 12B Problems/Problem 6
Solution 1
Looking at the polynomial, we can use Vieta's formulas. Let's say the roots are r, s, and t. Then
![\[r+s+t = -a\]](http://latex.artofproblemsolving.com/5/2/8/5285652525ea5dd27eb9050321e98bfc1d0eec48.png)
![\[rs+rt+st = b\]](http://latex.artofproblemsolving.com/f/8/4/f84cd6b1f09d7fb99f9c0ad730394cef05610b6f.png)
![\[rst = 6\]](http://latex.artofproblemsolving.com/4/6/f/46fa10a3403c50f0bfcd05efa7802e64b8ad9d4c.png)
We know that r, s, and t are integers, so we can list the possibilities for distinct solutions using the third formula. We find that 
can be 
and 
. Remember that r, s, and t have to be different. That's 5 solutions. We can also use those values to find the values of a and b to check that none of them are distinct, and we'll find that they are 
and 
. This means that there are 
