Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$.

x^2 - 5x  + t

The sum of the roots  =  5

We have  two possibilites.....  1,4   and  2, 3

So.....

(x - 1) ( x- 4)  =  x^2  -5x  + 4      →  t  = 4

And

(x - 2) (x -3)  = x^2  -5x + 6  → t = 6

Average value of t =  [ 4 + 6 ]  / 2  =  10/2   =  5