Question 6
Steve says to Jon, “I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3 - 2ax^2$ $+ (a^2 - 81)x - c$ for some positive integers $a$ and $c.$ Can you tell me the values of $a$ and $c?$”
After some calculations, Jon says, “There is more than one such polynomial.”
Steve says, “You’re right. Here is the value of $a.$” He writes down a positive integer and asks, “Can you tell me the value of $c?$”
Jon says, “There are still two possible values of $c.$”
Find the sum of the two possible values of $c.$
After some calculations, Jon says, “There is more than one such polynomial.”
Steve says, “You’re right. Here is the value of $a.$” He writes down a positive integer and asks, “Can you tell me the value of $c?$”
Jon says, “There are still two possible values of $c.$”
Find the sum of the two possible values of $c.$