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Exercise 3.2 · Q4

Q.Find a polynomial equation of minimum degree with rational coefficients, having 5−3\sqrt5-\sqrt3 as a root.

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Step 1. Identify all four required roots. Since 5,3\sqrt5,\sqrt3 are independent irrationals, rational coefficients force 5−3, 5+3, −5−3, −5+3\sqrt5-\sqrt3,\ \sqrt5+\sqrt3,\ -\sqrt5-\sqrt3,\ -\sqrt5+\sqrt3 to all be roots (Theorem 3.4).

Step 2. Pair them into two rational quadratic factors. (x−(5−3))(x−(5+3))=(x−5)2−3=x2−25x+2\big(x-(\sqrt5-\sqrt3)\big)\big(x-(\sqrt5+\sqrt3)\big)=(x-\sqrt5)^2-3=x^2-2\sqrt5x+2.

(x−(−5−3))(x−(−5+3))=(x+5)2−3=x2+25x+2\big(x-(-\sqrt5-\sqrt3)\big)\big(x-(-\sqrt5+\sqrt3)\big)=(x+\sqrt5)^2-3=x^2+2\sqrt5x+2. …

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