Question

divide x^2-4x+1 from x-2-√3

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Answer 1

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Answer 2

It is easy to see that x = 0 is the solution of the BPT (**) Consider x ≠ 0. Divide the two sides of the BPT for √x with the equivalent BPT: (√x + 1 / √x) + √ (x + 1 / x - 4) ≥ 3 (1) Let y = √x + 1 / √x ≥ 2 => y² = x + 1 / x + 2 => x + 1 / x - 4 = y² - 6 instead of (1) √ (y² - 6) ≥ 3 - y (2) If 2 ≤ y ≤ 3 then right side ≥ 0 should be squared (2) we have: y ≥ 5/2 or 5/2 ≤ y ≤ 3 If y> 3 then VP of (2) <0 should (2) evident The combination of (2) is y ≥ 5/2 <=> √x + 1 / √x ≥ 5/2 <=> 2 (√x) ² - 5√x + 2 ≥ 0 <=> √x ≤ 1/2 and √x ≥ 2 <=> 0 <x ≤ 1/4 and x ≥ 4 (***) KL: Combination (*); (**) and (***) BPT gave the solution 0 ≤ x ≤ 1 - √3 / 2 and x ≥ 4