primer-poly - A Primer on Polynomials : Polys Jonathan L.F....

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Unformatted text preview: A Primer on Polynomials : Polys Jonathan L.F. King University of Florida, Gainesville 32611-2082, USA squash@math.ufl.edu Webpage http://www.math.ufl.edu/ squash/ 2 February, 2010 (at 13:33 ) Preliminaries. An expression such as k N ( read as k is an element of N or k in N ) means that k is a natural number; a natnum . N = naturalnumbers = { , 1 , 2 ,... } . Z = integers = { ...,- 2 ,- 1 , , 1 ,... } . For the set { 1 , 2 , 3 ,... } of positive integers, the posints , use Z + . Use Z- for the negative integers, the negints . Q = rationalnumbers = { p q | p Z and q Z + } . Use Q + for the positive ratnums and Q- for the negative ratnums. R = reals. The posreals R + and the negreals R- . C = complex numbers, also called the complexes . Abbrevs. Poly(s) : polynomial(s) . Irred : irreducible . Coeff : coefficient and var(s) : variable(s) and parm(s) : parameter(s) . Expr. : expression . Fnc : function (so ratfnc : means rational function , a ratio of polynomials). Seq : sequence . Soln : solution . Propn : proposition . CEX : Counter- example . Eqn : equation . RhS: RightHand Side of an eqn or inequality. LhS: lefthand side . Sqrt : square-root , e.g, the sqrt of 16 is 4. Cts : continuous and cty : continuity . Ptn : partition , but pt : point , as in a fixed-pt of a map. Below we will view various expressions as fncs of a vari- able, x . As usual, x is a another name for the constant fnc 1. Outline Preliminaries . . . . . . . . . . . . . . . . . . . 1 Monomials . . . . . . . . . . . . . . . . . . . . 1 Polynomials . . . . . . . . . . . . . . . . . . . . 1 Degree . . . . . . . . . . . . . . . . . . . . . . . 2 Upper-bounding degree . . . . . . . . . . . . . 2 The zeros of a polynomial . . . . . . . . . . . . . . . . 2 Dont Panic ! . . . . . . . . . . . . . . . . . . 3 Factoring . . . . . . . . . . . . . . . . . . . . . 3 Zeros/Roots of fncs . . . . . . . . . . . . . . . 3 The Quadratic Formula ( QF ) . . . . . . . . . . . . . . 3 Irreducibility and the QF . . . . . . . . . . . . 4 Fully factored form . . . . . . . . . . . . . . . . 4 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . 4 Division . . . . . . . . . . . . . . . . . . . . . . 7 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 8 Algebraic and Transcendental numbers . . . . . . . . . . . . 8 Subfields of C . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Quadratic extensions . . . . . . . . . . . . . . . . . . . 9 Case: S and P are integers . . . . . . . . . . . 10 More general coefficients . . . . . . . . . . . . . . . . . . . . 11 A Appendices 11 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . 11 Making | z | small . . . . . . . . . . . . . . . . . 12 Integrating polynomials . . . . . . . . . . . . . . . . . . . . 12 Bernstein polynomials . . . . . . . . . . . . . .Bernstein polynomials ....
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primer-poly - A Primer on Polynomials : Polys Jonathan L.F....

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