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Unformatted text preview: CSE 1400 Applied Discrete Mathematics Polynomials Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Polynomials 1 Constants 1 Linear Polynomials 2 Quadratic Polynomials 2 The Power Basis 3 The Binomial Theorem 4 Horners Rule for Evaluating Polynomials 4 Taylor Polynomials 5 Falling Factorial Powers 5 Rising Factorial Powers 7 Problems on Polynomials 8 Abstract Polynomial functions are computationally primitive and form bases for approximating more complicated functions. Polynomials Polynomials are important because 1 . They are easy to evaluate. 2 . They can approximate arbitrarily well many more complex func tions. If f is a continuous realvalued function on [ a , b ] and if any e > 0 is given, then there exists a polynomial p on [ a , b ] such that  f ( x ) p ( x )  < e for all x in [ a , b ] . In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy. Low degree polynomials p ( x ) are studied in college algebra. College Instances of polynomials include p ( x ) = 3 x + 2 p ( x ) = x 2 x 1 p ( x ) = x 2 4 p ( x ) = x 3 + x 2 + x + 1 algebra studies how to solve polynomial equations p ( x ) = 0. and The zeros of polynomials can be com puted. 3 x + 2 = at x = 2/3 x 2 x 1 = at x = 1 5 2 x 2 4 = at x = 2 x 3 + x 2 + x + 1 = at x = 1, i students learn to graph polynomial equations p ( x ) = y in a Cartesian coordinate system. cse 1400 applied discrete mathematics polynomials 2 Constants C onstants are polynomials . Constants are polynomials of de gree . There are famous, important constants. , Zero 1 , One 2 1.414213, The square root of 2 = ( 1 + 5 ) /2 1.618033, The golden ratio 3.141592, pi e 2.718281, Euler s or Napiers constant 0.577215, The EulerMascheroni constant Linear Polynomials P olynomials that grow by a constant increment are called linear . A linear polynomial p ( x ) is written The zero of the equation p ( x ) = 0 is x = m 1 b . p ( x ) = mx + b for some slope m R , m 6 = 0 and y intercept b R . These functions are called first degree polynomials....
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This note was uploaded on 02/11/2012 for the course MTH 2051 taught by Professor Shoaff during the Fall '11 term at FIT.
 Fall '11
 Shoaff
 Math, Binomial Theorem, Polynomials, Binomial

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