Given that the polynomial \(x^2 - kx + 60\) has only positive integer roots, find the average of all distinct possibilities for k.
I hope this is correct.
k can be 61, 32, 23, 19, 17, or 16. When we add and divide by 6, we get 28.
+9674 By Vieta's formula, the product of roots of \(x^2 - kx + 60\) is 60.
What integers multiply to 60?
| 1 | 60 |
| 2 | 30 |
| 3 | 20 |
| 4 | 15 |
| 5 | 12 |
| 6 | 10 |
| 10 | 6 |
| 12 | 5 |
| 15 | 4 |
| 20 | 3 |
| 30 | 2 |
| 60 | 1 |
Each row represents the two roots of a specific value of k.
The sum of roots is k by Vieta's formula again. Therefore, you can find the possibilities for the sum of roots, then find the average.
