By Bor-Luh Lin

This quantity includes the court cases from a learn Workshop on Banach house concept held on the collage of Iowa in Iowa urban in July 1987. The workshop supplied individuals with a collaborative operating surroundings within which principles should be exchanged informally. a number of papers have been initiated through the workshop and are awarded right here of their ultimate shape. additionally incorporated are contributions from numerous specialists who have been not able to wait the workshop. not one of the papers may be released somewhere else. through the workshop, hours on a daily basis have been dedicated to seminars on present difficulties in such components as susceptible Hilbert areas, zonoids, analytic martingales, and operator concept, and those subject matters are mirrored in the various papers within the assortment

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The sum of polynomials being a polynomial, the result follows. 17. The remainder theorem Theorem. f(c). If a polynomial f(x) is divided by x − c, then the remainder is 23 ALGEBRA AND TRIGONOMETRY Proof. Divide x into f(x), getting f( ). Multiply f( ) times x − c to get f(x) − f(c). Subtracting, we get f(c). Symbolically f( ) x - c f ( x) f ( x ) - f (c ) f (c) Contributed by Richard Laatsch of Miami University in Oxford, OH. ♥ 18. The zero polynomial Two colleagues were examining this proof that only the zero polynomial could take the value zero everywhere.

Now construct a trapezoid DEF G which is the upper part of a regular hexagon of side d. Thus DG = 2DE = 2EF = 2F G = 2d. The area of the semi-circle with diameter DG is four times the area S of the semi-circle of diameter d constructed on each of the sides DE, EF , F G as diameter. It can be seen that the area S plus the area of the three lunes (L) is equal to the area of the trapezoid (T ). 2 F GEOMETRY 41 Symbolically, we have R = 2L and T = 3L + S. Hence the area of the given circle is 2S = 2T − 6L = 2T − 3R.

4 4 Hence p 3 =− y2 4 and q 1 = − cos 3θ, y3 4 so that y2 = −(4p/3) and cos 3θ = −(4q/y3 ) determine y and θ. For example, trying this on x3 − 3x − 1 = 0 yields y = 2, cos 3θ = 1/2 or 3θ ≡ π/3, 5π/3 (mod 2π). 87938 . . 34729 . . 53209 . . ) Is this method on the level? ♣ x = 2 cos ALGEBRA AND TRIGONOMETRY 19 The justification can be given along these lines. The substitution x = y cos θ introduces an extra degree of freedom. Accordingly, we can select y so that p/y2 = −3/4; thus, θ must satisfy the conditional equation cos3 θ + (−3/4) cos θ + yq3 = 0.