The roots of the polynomial x^3 - 52x^2 + 581x - k are distinct prime numbers. Find k.

Guest Aug 11, 2019

#1**+1 **

\(((x-a)(x-b)(x-c) =x^3 + x^2 (-a-b-c)+x (a b+a c+b c)-a b c \\ 52=a+b+c\\ 581 = ab + ac + bc \\ k = abc\)

\(\text{Solving this is a pain but conceptually straightforward.}\\ \text{Start by knowing one of them must be 2, as the sum of 3 odds is odd and 52 is even}\\ \text{The roots are $2, ~13,~37$}\\ k = 2 \cdot 13 \cdot 37 = 962\)

.Rom Aug 11, 2019