Given that the polynomial \(x^2 - kx + 60\) has only positive integer roots, find the average of all distinct possibilities for k.

Guest May 3, 2022

#1**0 **

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**.

Guest May 3, 2022

#2**0 **

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.

MaxWong May 3, 2022