Complex numbers and polynomials - University College Cork

Prove that b is a positive integer if, and only if, a is a positive integer of theform 1 2 n(n2 + 3), for some positive integer n.By direct computation it can be seen that b is a root of the cubic x 3 + 3x − 2a.Hence, if b is a positive integer, then so is 2a = b 3 + 3b. But the latter is even.Hence, a is a positive integer. Conversely, if a is of the form 1 2 n(n2 + 3), forsome positive integer n, then n is also a root of the same cubic. But this cubichas only one real root. Thus b = n.4. IRMO95. Suppose that a, b and c are complex numbers, and that all threeroots z of the equationx 3 + ax 2 + bx + c = 0satisfy |z| = 1 (where | | denotes absolute value). Prove that all three roots wof the equationx 3 + |a|x 2 + |b|x + |c| = 0also satisfy |w| = 1.Denote the roots of x 3 + ax 2 + bx + c = 0 by p, q, r. ThenHence |c| = 1, andThus−a = p + q + r, b = pq + qr + rp, −c = pqr.|b| = |(pqr)( 1 r + 1 p + 1 )| = |¯r + ¯p + ¯p| = |a|.qx 3 + |a|x 2 + |b|x + |c| = x 3 + |a|(x 2 + x) + 1 = (x + 1)(x 2 + (|a| − 1)x + 1);and (?) since ||a| − 1| ≤ 2, the roots of the quadratic x 2 + (|a| − 1)x + 1 arecomplex numbers of unit modulus, the result follows.5. Prove that x 4 + x 3 + x 2 + x + 1 is a factor of x 44 + x 33 + x 22 + x 11 + 1.Since x 5 − 1 = (x − 1)(x 4 + x 3 + x 2 + x + 1) = 0 we see that x 5n = 1 for everyinteger n. So x 44 + x 33 + x 22 + x 11 + 1 = x 8.5 x 4 + x 6.5 x 3 + x 4.5 x 2 + x 2.5 x + 1 =x 4 + x 3 + x 2 + x + 1 = 0. Thus every root of x 4 + x 3 + x 2 + x + 1 is a root ofx 44 + x 33 + x 22 + x 11 + 1, and the Remainder Theorem does the rest.6. The coeffs of the cubic ax 3 + bx 2 + cx + d are integers, ad is odd and bc is even.Prove that at least one root is irrational.Suppose all roots are rational, and apply the rational roots theorem. Accordingto this, if p/q is a rational root with p, q in their lowest form, then p divides31