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# Vieta's formula problem

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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