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

 Aug 11, 2019
 #1
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\(((x-a)(x-b)(x-c) =x^3 +  x^2 (-a-b-c)+x (a b+a c+b c)-a b c \\ 52=a+b+c\\ 581 = ab + ac + bc \\ k = abc\)

 

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

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 Aug 11, 2019
 #2
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You could also just use the rational root theorem

 Aug 11, 2019

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