Question 20

The value of $$k$$ for which the equation $$x^3 - 6x^2+11x+(6-k)=0$$ has exactly three positive integer solutions is ------.

Correct Answer: 12

Solution

The sum of the integer roots is $$6$$ and the pairwise-product sum is $$11$$, the only positive integer triple that satisfies this condition is $$1, 2, 3$$, since $$1+2+3=6$$ and $$1\cdot2+2\cdot3+3\cdot1=11$$.

The product of roots of the cubic can be found as $$k-6$$

Comparing with $$1\times2\times3=6$$ we get $$k=12$$.

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