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0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 } {PSTYLE "_pstyle8" -1 212 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 0 0 0 8 4 1 0 1 0 2 2 -1 1 }{CSTYLE "_cstyle7" -1 210 " Times" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle9" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle10" -1 214 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 205 "" 0 "" {TEXT 204 27 "Computing with Polynomi als " }{TEXT 204 0 "" }}{PARA 206 "" 0 "" {TEXT 205 0 "" }}{PARA 207 " " 0 "" {TEXT 206 15 "Mike O'Sullivan" }{TEXT 206 0 "" }}{PARA 208 "" 0 "" {TEXT 207 26 "San Diego State University" }{TEXT 207 0 "" }}{PARA 209 "" 0 "" {TEXT 208 12 "January 2005" }{TEXT 208 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 84 "This worksheet has some of the basic c omputations that can be done with polynomials." }{TEXT 209 0 "" } {TEXT 209 46 "\nSee Help:Polynomials:entering for more info." }{TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 76 "You may also want to explor e two packages, PolynomialTools[] and Grobner[]. " }{TEXT 209 0 "" }}} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 210 22 "Univariate Polynomials" } {TEXT 210 0 "" }}{PARA 210 "" 0 "" {TEXT 209 65 "A nice property of Ma ple is that it can do symbolic computation. " }{TEXT 209 0 "" }} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 19 "f := x^2 + 7*x + 3;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 12 "\ng := f^2-9;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 11 "\nexpand(g);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 16 "degree(g); " }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 56 "\nldegree(g); #the degree of the lowest degree t erm." }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 54 "\nirreduc(f); #is f irreducible over the ra tionals?" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 17 "\nsolve(f); " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 169 "The proce dure factors() gives the unique factorization as a list. with the lead ing coefficient, and the monic irreducible factors, each of these in [ factor, power] from." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 10 "factor(g);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 18 "\nFg := factors(g);" }{MPLTEXT 1 0 74 "\nFg[2,1,1]; Fg[2,1,2]; #The fir st factor in the list, and its exponent." }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 82 "Notice what happens when the polynomial is of large degr ee and we try to solve it." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 9 "solve(g);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 12 "\n solve(g+1);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 14 "\nsolve(x*g+1);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 36 "To evaluat e a polynomial use subs()." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 22 "h := sum(x^i, i=1..8);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 15 "\nsubs( x=1, h);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 15 "\nsubs(x=-1, h);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 44 "\nsubs(x= I , h); #I is the square root of -1." }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 11 "\nfactor(h);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 73 "Here we convert to a form that is computationally effic ient to evaluate." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 19 "convert(h, horner);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 105 "This computes the gcd and makes r and s hold the multiples that give the gcd (over the rational numbers)." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 25 "gcdex(f , h, x, 'r', 's');" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 6 "\nr; s;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 10 "\nr*f+ s*h;" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 55 "Unfortun ately you often have to tell Maple to simplify." }{TEXT 209 0 "" }}} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 20 "simplify(r*f + s*h);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 71 "When h has a multiple root, h and h' have that root as a common factor." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 41 "h := x^5- 3*x^ 4 + 2* x^3 - 6*x^2 + x - 3;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 11 "\nfa ctor(h);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 18 "\nhp := diff(h, x);" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 13 "\ngcd(h, hp); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 211 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 210 24 "Multivariate polynom ials" }{TEXT 210 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 209 86 "A polyn omial in several variables may also be considered as a polynomial in o ne of the" }{TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 78 "variables with coefficients in the ring of polynomials in the other variables." }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 33 "F := ( x^2+1)*(y+7) + y^2*(x+1)^3;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 17 "\nF1 := expand(F);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 46 "\nF2 := collect(F , x); #F as a polynomial in x" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 45 " \nF3 := collect(F, y); #F as a polynomial in y" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 209 85 "The degree and leading coeffici ent depend on whether you consider F as a polynomial " }{TEXT 209 0 " " }}{PARA 210 "" 0 "" {TEXT 209 102 "in two variables, or as a polynom ial in one of the variables with coefficients in the polynomial ring " }{TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 13 "of the other." } {TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 11 "degree(F 1);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 12 "\nlcoeff(F1);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 15 "\ndegree(F1, x);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 15 "\nlcoeff(F1, x);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 15 "\ndegree(F1, y);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 15 "\nlcoeff(F1 , y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 67 " T he sort function uses tdeg (total degree lexicographic order) or " } {TEXT 209 0 "" }{TEXT 209 69 "\nplex (pure lexicographic order). You \+ can specify whether x " 0 "" {MPLTEXT 1 0 22 "sort(F1, [y,x], plex);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 23 "\nsort(F1, [x,y], plex);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 23 "\nsort(F1, [y,x], tdeg);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 23 "\nsort(F1, [x,y], tdeg);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 63 "You can \+ compute the gcd with respect to one of the variables. " }{TEXT 209 0 "" }{TEXT 209 102 "\nNotice that the coefficients r and s are in the f ield of rational polynomials in the other variable. " }{TEXT 209 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 32 "f := x^2*y+y*x; g := x ^2+x*y+x;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 25 "\ngcdex(f,g, y,'r', 's '); " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nr; s; " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 49 "\nsimplify(r*f + s*g); #You can also use normal." }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 59 "\n #I'm no t sure about the difference." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 211 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 210 31 "Univariate Rational Polynomials" } {TEXT 210 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 209 60 "Here is the ra tional polynomial r from the previous section." }{TEXT 209 0 "" }}} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 2 "r;" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 10 "\nnumer(r);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 10 "\nd enom(r);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 201 30 "Convert to a partial fraction." }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 23 "convert(r, parfrac, x);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 214 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }