Complex numbers and polynomials - University College Cork

8. Let a, b, c and d be real numbers with a ≠ 0. Prove that if all the roots of thecubic equationaz 3 + bz 2 + cz + d = 0lie to the left of the imaginary axis in the complex plane, thenab > 0, bc − ad > 0, ad > 0.9. IRMO93. The real numbers α, β satisfy the equationsα 3 − 3α 2 + 5α − 17 = 0,Find α + β.β 3 − 3β 2 + 5β + 11 = 0.10. IRMO93. Let a 0 , a 1 , . . . , a n−1 be real numbers, where n ≥ 1, and let thepolynomialf(x) = x n + a n−1 x n−1 + . . . + a 0be such that |f(0)| = f(1) and each root α of f is real and satisfies 0 < α < 1.Prove that the product of the roots does not exceed 1/2 n .11. IRMO93. Let a 1 , a 2 . . . a n , b 1 , b 2 . . . b n be 2n real numbers, where a 1 , a 2 . . . a nare distinct, and suppose that there exists a real number α such that theproduct(a i + b 1 )(a i + b 2 ) . . . (a i + b n )has the value α for i = 1, 2, . . . , n. Prove that there exists a real number βsuch that the producthas the value β for j = 1, 2, . . . , n.(a 1 + b j )(a 2 + b j ) . . . (a n + b j )12. IRMO94. Determine, with proof, all real polynomials f satisfying the equationfor all real numbers x.f(x 2 ) = f(x)f(x − 1),24