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Polynomials - CSE 1400 Applied Discrete Mathematics...

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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 Horner’s 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 real-valued 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 = 0 at x = - 2/3 x 2 - x - 1 = 0 at x = 1 ± 5 2 x 2 - 4 = 0 at x = ± 2 x 3 + x 2 + x + 1 = 0 at x = - 1, ± i students learn to graph polynomial equations p ( x ) = y in a Cartesian coordinate system.
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cse 1400 applied discrete mathematics polynomials 2 Constants C onstants are polynomials . Constants are polynomials of de- gree 0 . There are famous, important constants. 0 , Zero 1 , One 2 1.414213, The square root of 2 ϕ = ( 1 + 5 ) /2 1.618033, The golden ratio π 3.141592, pi e 2.718281, Euler ’s or Napier’s constant γ 0.577215, The Euler-Mascheroni 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. Lines through the origin are interesting. They have the form ax + by = 0 where ( a 6 = 0 ) ( b 6 = 0 ) One-dimensional affine space establishes an equivalence between ax + by is the inner product of two vectors h a , b i and h x , y i . When the inner product is 0 , the vectors are orthogonal (perpendicular).
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