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Question 17

The equation $$x^4 - 4x^3 + ax^2 + bx + 1 = 0$$ has 4 positive roots. Then $$a + b$$ is


Correct Answer: 2

Solution

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The four positive roots have sum 4 and product 1, so their arithmetic mean is 1 and their geometric mean is also 1. 

Equality in the AM-GM inequality forces all four roots to equal 1, so the equation is $$(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 = 0$$. 

Hence $$a = 6$$, $$b = -4$$ and $$a + b = 2$$.

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