### Instruction: Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$.### Response: I notice that this is a quadratic polynomial, so I can use the fact that the sum of the roots is equal to the opposite of the coefficient of $x$, and the product of the roots is equal to the constant term.
That means, if $r$ and $s$ are the roots, then $r+s=5$ and $rs=t$.
Since the roots are positive integers, I can list all the possible pairs of $r$ and $s$ that add up to 5: (1,4), (2,3), (3,2), and (4,1).
For each pair, I can find the corresponding value of $t$ by multiplying the roots: 1*4=4, 2*3=6, 3*2=6, and 4*1=4.
So the possible values of $t$ are 4 and 6, and they are distinct.
To find the average of these values, I can add them up and divide by the number of values: (4+6)/2=5.