Complex numbers and polynomials - University College Cork

with equality iff α = β = γ. Thus p ′ has two real roots, which may be equal. □Theorem 8. Suppose p is a cubic, not necessarily with real coefficients, andThenProof. SupposeThenHenceBut also,which means thatp ′ (α) = p ′ (β) = 0 = p ′′ (γ).γ = α + β .2p(x) = ax 3 + bx 2 + cx + d, a ≠ 0.p ′ (x) = 3ax 2 + 2bx + c, p ′′ (x) = 6ax + 2b.γ = −b3a .p ′ (x) = 3a(x − α)(x − β) = 3a(x 2 − (α + β)x + αβ),−3a(α + β) = 2b, α + β = −2b3a = 2γ.□This means the if a cubic with real coefficients has two turning points, then themid-point of their first coordinates coincides with the point of inflexion.6.2 Roots of cubicsTo discuss the roots of a general polynomial with real coefficients, we need to compilesome preliminary facts. We start with an identity which is useful in other situationsas well.Lemma 1. Suppose x, y, z are real or complex numbers. Thenx 3 + y 3 + z 3 − 3xyz = (x + y + z)(x 2 + y 2 + z 2 − xy − yz − zx).Proof. Simply expand the RHS. □Taking y = 1, z = 0 we get the more familiar factorisation:x 3 + 1 = (x + 1)(x 2 − x + 1),while the choice y = −1, z = 0 tells us thatx 3 − 1 = (x − 1)(x 2 + x + 1).15