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Unformatted text preview: a polynomial and provide a mechanism for extracting coeﬃcients. > poly := 3*z^3 - z^2 + 2*z - 3*z + 1; 3.4 Polynomials Table 3.1 Commands for Finding Polynomial Coeﬃcients • 63 Command coeff lcoeff tcoeff coeffs degree ldegree Description extract coeﬃcient ﬁnd the leading coeﬃcient ﬁnd the trailing coeﬃcient return a sequence of all the coeﬃcients determine the (highest) degree of the polynomial determine the lowest degree of the polynomial poly := 3 z 3 − z 2 − z + 1 > coeff(poly, z^2); −1 > degree(poly,z); 3 Root Finding and Factorization The solve command determines the roots of a polynomial whereas the factor command expresses the polynomial in fully factored form. > poly1 := x^6 - x^5 - 9*x^4 + x^3 + 20*x^2 + 12*x; poly1 := x6 − x5 − 9 x4 + x3 + 20 x2 + 12 x > factor(poly1); x (x − 2) (x − 3) (x + 2) (x + 1)2 > poly2 := (x + 3); poly2 := x + 3 > poly3 := expand(poly2^6); 64 • Chapter 3: Finding Solutions Table 3.2 Functions that Act on Polynomials Function content compoly discrim gcd gcdex interp lcm norm prem primpart randpoly recipoly resultant roots sqrfree Description content of a multivariate polynomial polynomial decomposition discriminant of a polynomial greatest common divisor extended Euclidean algorithm polynomial interpolation least common multiple norm of a polynomial pseudo-remainder primitive part of a multivariate polynomial random polynomial reciprocal polynomial resultant of two polynomials roots over an algebraic number ﬁeld square-free factorization poly3 := x6 + 18 x5 + 135 x4 + 540 x3 + 1215 x2 + 1458 x + 729 > factor(poly3); (x + 3)6 > solve({poly3=0}, {x}); {x = −3}, {x = −3}, {x = −3}, {x = −3}, {x = −3}, {x = −3} > factor(x^3 + y^3); (x + y ) (x2 − x y + y 2 ) Maple factors the polynomial over the ring implied by the coeﬃcients, for example, the integers or rational numbers. The factor command also allows you to specify an algebraic number ﬁeld over which to factor the polynomial. For more information, refer to the ?factor help page. For a list of functions that act on polynomials, see Table 3.2. 3.5 Calculus • 65 3.5 Calculus Maple provides many powerful tools for solving problems in calculus. This section presents the following concepts. • Computing the limits of functions by using the Limit command • Creating series approximations of a function by using the series command • Symbolically computing derivatives, indeﬁnite integrals, and deﬁnite integrals About Calculus Examples in This Manual This and the following section provide an introduction to the Maple commands for calculus problems. A more extensive look at the Student[Calculus] package and the calculus commands in the main library is provided in chapter 4 and chapter 7. Also, detailed information is available in the Maple help system. Using the Limit Command Compute the limit of a rational function as x approaches 1. > f := x -> (x^2-2*x+1)/(x^4 + 3*x^3 - 7*x^2 + x+2); f := x → > Limit(f(x), x=1); x2 − 2 x + 1 x4 + 3 x3 − 7 x2 +...
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## This note was uploaded on 08/27/2012 for the course MATH 1100 taught by Professor Nil during the Spring '12 term at National University of Singapore.

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