Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

Find an ordered pair (a,b)(a, b) of real numbers for which x2+ax+bx^{2}+a x+b has a non-real root whose cube is 343.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The cube roots of 343 are the roots of x3−343x^{3}-343, which is (x−7)(x2+7x+49)(x-7)\left(x^{2}+7 x+49\right). Therefore the ordered pair we want is (7,49)(\mathbf{7, 49}).

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