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primer-poly - Quadratic extensions Case S and P are...

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A Primer on Polynomials : Polys Jonathan L.F. King University of Florida, Gainesville 32611-2082, USA [email protected] 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 = natural numbers = { 0 , 1 , 2 , . . . } . Z = integers = { . . . , - 2 , - 1 , 0 , 1 , . . . } . For the set { 1 , 2 , 3 , . . . } of positive integers, the posints , use Z + . Use Z - for the negative integers, the negints . Q = rational numbers = { 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 ’. Prop’n : 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 0 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 Don’t 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 . . . . . . . . . . . . . . 12 Integration with convolutions . . . . . . . . . . 13 Polynomials in several variables . . . . . . . . . . . . . . . . 14 Roots of the Cubic and the Quartic . . . . . . . . . . . . . 14 Sum and product . . . . . . . . . . . . . . . . . 14 Roots of the depressed cubic . . . . . . . . . . . . . . . 14 Cardano’s formula . . . . . . . . . . . . . . . . 15 General cubic . . . . . . . . . . . . . . . . . . . 15 Roots of the Quartic . . . . . . . . . . . . . . . . . . . 15 Ferrari’s formula . . . . . . . . . . . . . . . . . 16 Index for “Primer on Polynomials” 16 Monomials. Here are examples: - 6 x 2 , 7 · x 10 , π x, 3 , 0 . 1a : Here are non- examples: x , e x , ln( x ) , sin( x ) . 1b : A monomial is an expression OTForm Bx n , where n N and B is a number ( in R or C ), called the coefficient of x n . To justify the monomials of (1a), note x 5 = 1 · x 5 , 3 = 3 · x 0 and 0 = 0 · x 0 . In contrast, the expressions in (1b) don’t look like monomi- als, although it would take some wrestling to show, for example, that 3 x note === x 1 / 3 does not equal some Bx n . It is easy to show that e x is not a monomial: e x has a horizontal asymp- tote as x & , yet the only monomials with a horiz. asymptote are the constants. And e x is not constant.
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