Complex numbers and polynomials - University College Cork

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Example 2. The pairs of real numbers (m, c) such that y = mx + c is a line belowthe graph of the square function comprise the set{(m, c) : m 2 + 4c ≤ 0}.Proof. Suppose (m, c) generates such a line. Thenmx + c ≤ x 2 , ∀x ∈ R.Equivalently, the quadratic polynomial x 2 − mx − c is nonnegative for all real x. Bythe previous example, this occurs iff m 2 ≤ 4(−c). The result follows. □4 Exercises1. Sketch the graphs of the polynomials−3x+2, 2x−3, −3x 2 +4x−2, 3x 2 +4x−2, (x−α)(x−β), −(x−α)(x−β),where α, β are arbitrary real numbers.2. Determine the minimum of each of the quadratics(x − 1)(x − 2), (x − 3)(x − 4), (x − 1)(x − 3), (x − 2)(x − 4).3. Determine the minimum of the quadratic(x − 1) 2 + (x − 2) 2 + (x − 3) 2 + (x − 4) 2 .4. Determine the minimum of the quadratica(x − α) 2 + b(x − β) 2 + c(x − γ) 2 ,where a, b, c, α, β, γ are arbitrary real numbers, and at last one of a, b, c isnon-zero.5. Determine the minimum of each of the quartics(x−1)(x−2)(x−3)(x−4), (x−1)(x−2)+(x−1)(x−2)(x−3)(x−4)+(x−3)(x−4).6. Let P = (1, −1), Q = (−1, 1). Show that there is a point R on the line L,whose equation is x + y = 1, such that|P R| 2 + |RQ| 2 ≤ |P X| 2 + |XQ| 2 , ∀X ∈ L.10
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
Page 1 and 2: Enrichment Lectures 2010Some facts
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Page 5 and 6: 3 Linear and quadratic polynomialsH
Page 7 and 8: 3.3 Heron’s problemGiven two dist
Page 9: We can also conclude that the graph
Page 13 and 14: 2. Let p be a arbitrary quartic. Us
Page 15 and 16: with equality iff α = β = γ. Thu
Page 17 and 18: Plainly, the pair a = 1, b = 2 work
Page 19 and 20: Note especially that if a is ration
Page 21 and 22: and, for k ≥ 2,c k =Again,( mk)b
Page 23 and 24: 2. For which n ∈ N is x 2 +x+1 a
Page 25 and 26: 13. IRMO95. Suppose that a, b and c
Page 27 and 28: q = −3uv, r = u 3 + v 3 for u, v.
Page 29 and 30: To utilize this, replace x by ρ co
Page 31 and 32: Prove that b is a positive integer

Complex numbers and polynomials - University College Cork

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