Challenge Basic Math Problem of the Week 10/25/2017

[PF PotW Robot]

Here is this week's basic math problem. We have several members who will check solutions, but we also welcome the community in general to step in. We also encourage finding different methods to the solution. If one has been found, see if there is another way. Using spoiler tags is optional. Occasionally there will be prizes for extraordinary or clever methods. Spoiler tags are optional.

Find all real numbers $k$ that give the three roots of the cubic equation $5x^3-5(k+1)x^2+(71k-1)x-(66k-1)=0$ are positive integers.

(PotW thanks to our friends at http://www.mathhelpboards.com/)

 

[Buzz Bloom]

Find all real numbers k that give the three roots of the cubic equation 5x3−5(k+1)x2+(71k−1)x−(66k−1)=05x^3-5(k+1)x^2+(71k-1)x-(66k-1)=0 are positive integers.

This problem statement does not parse into understandable English for me. Here is my guess about what it means.

The roots of the cubic equation are to be integers, and this is an additional constraint on the acceptable values of k besides that k must be a real number. Is this correct?

 

[mfb]

That’s how I interpreted it as well.

“Find all real numbers k such that ... has three positive integers as roots.”

 

[MarkFL]

Hello, PF community!

I was contacted by Greg, who let me know there was some confusion regarding this problem.

I posted that problem over at MHB, but I was standing in for our regular Secondary School POTW Director, and she provided the problem to me to post in her absence.

What you want to find is all values of $k\in\mathbb{R}$ such that the given cubic polynomial will have positive integers for all three of its roots.