Complex numbers and polynomials - University College Cork

7. Let P, Q be two points not necessarily on the same side of a line L. Prove thatthere is a unique point R ∈ L such that|P R| 2 + |RQ| 2 ≤ |P X| 2 + |XQ| 2 , ∀X ∈ L.8. Let x, y be a pair of real numbers, prove thatwith equality iff x = y = 1.x 2 + y 2 + 2 ≥ (x + 1)(y + 1),9. Show that the set {(x, y) ∈ R 2 : y ≥ |x|} is convex, and describe all the pairs(m, c) such that the line y = mx + c lies below the graph of y = |x|.10. Show that the semicircle {(x, y) : −1 ≤ x ≤ 1, y = − √ 1 − x 2 } is convex, anddescribe all the pairs (m, c) such that the line y = mx + c lies below its graph.11. For what real numbers a ≠ 0, b, c is the quadratic ax 2 + bx + c nonnegative onthe half-line [0, ∞)? On the interval [0, 1]?5 Quartic polynomialsA quartic is a polynomial of degree 4, i.e., a linear combination of the simple monomials1, x, x 2 , x 3 , x 4 , and is therefore of the formp(x) = ax 4 + bx 3 + cx 2 + dx + e,where the coefficients a, b, c, d, e are real or complex numbers, and a ≠ 0.Every quartic is a product of two quadratics. For, counting their multiplicity, sucha polynomial has 4 roots, and if these are denoted by α, β, γ, δ, thenp(x) = a(x − α)(x − β)(x − γ)(x − δ) = a(x 2 − (α + β)x + αβ)(x 2 − (γ + δ)x + γδ),a product of two quadratics, which evidently are not unique. Conversely, it’s easyto see that the product of two quadratics is a quartic.Theorem 5. If a quartic has real coefficients, then it can be expressed as a productof two quadratics each having real coefficients.Proof. Supposep(x) = ax 4 + bx 3 + cx 2 + dx + e,where a, b, c, d, e are real, and a ≠ 0. The result is immediate if p has only realroots. If z is a complex root of p, then so is ¯z, by Theorem 3. Since p has at most4 distinct roots, either they are all complex, and come in pairs, or at most two arenon-real. If they are all complex, then we can write them as α, ᾱ, β, ¯β, in which casep(x) = a(x 2 −(α+ᾱ)x+αᾱ)(x 2 −(β+ ¯β)x+β ¯β) = a(x 2 −(α+ᾱ)x+|α| 2 )(x 2 −(β+ ¯β)x+|β| 2 ).11