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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "This worksheet builds extensions of the i
ntegers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Polynomials
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "The really nice thing about M
aple is that it can do symbolic computation.  That means that it not o
nly computes numbers but can also compute with polynomials. Here we se
e just a touch of Maple's capability." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Define two polynomials." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x^2+x+1;  g:= x+3;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$)%\"xG\"\"#\"\"\"F*F(F*F*F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&%\"xG\"\"\"\"\"$F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Compute their product and sum." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h := f*g; k := f+g;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*&,(*$)%\"xG\"\"#\"\"\"F+F)F+F+F
+F+,&F)F+\"\"$F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,(*$)%\"xG
\"\"#\"\"\"F**&F)F*F(F*F*\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "hh:= expand(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#hhG,**$)%\"xG\"\"$\"\"\"F**&\"\"%F*)F(\"\"#F*F**&F,F*F(F*F*F)F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(hh);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"F)F'F)F)F)F),&F'F)\"\"$F)F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "degree(h); degree(hh);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "We can a
lso get the coefficient of x^i in h.  \nFor some reason coeffs(h,x); y
ields an error, while coeffs(hh,x); does not." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "coeff(h, x,
 1);\ncoeff(h,x,0);\ncoeffs(h,x);\ncoeffs(hh,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}
{PARA 8 "" 1 "" {TEXT -1 35 "Error, invalid arguments to coeffs\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\"\"%F$\"\"\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 42 "Here you see some of the power of Maple.  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "factor(x^100-1);\nffac:= fac
tors(x^100-1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*4,&%\"xG\"\"\"F&!\"
\"F&,,*$)F%\"\"%F&F&*$)F%\"\"$F&F&*$)F%\"\"#F&F&F%F&F&F&F&,,*$)F%\"#?F
&F&*$)F%\"#:F&F&*$)F%\"#5F&F&*$)F%\"\"&F&F&F&F&F&,&F%F&F&F&F&,,F&F&F%F
'F/F&F,F'F)F&F&,,F&F&F<F'F9F&F6F'F3F&F&,&F&F&F/F&F&,,*$)F%\"\")F&F&*$)
F%\"\"'F&F'F)F&F/F'F&F&F&,,*$)F%\"#SF&F&*$)F%\"#IF&F'F3F&F9F'F&F&F&" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ffacG7$\"\"\"7+7$,&%\"xGF&F&!\"\"F
&7$,,F&F&F*F+*$)F*\"\"#F&F&*$)F*\"\"$F&F+*$)F*\"\"%F&F&F&7$,,F&F&*$)F*
\"\"&F&F+*$)F*\"#5F&F&*$)F*\"#:F&F+*$)F*\"#?F&F&F&7$,&F&F&F.F&F&7$,,FB
F&F?F&F<F&F9F&F&F&F&7$,,*$)F*\"#SF&F&*$)F*\"#IF&F+FBF&F<F+F&F&F&7$,,*$
)F*\"\")F&F&*$)F*\"\"'F&F+F4F&F.F+F&F&F&7$,,F4F&F1F&F.F&F*F&F&F&F&7$,&
F*F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "In the output of i
factors(),  the first entry, 1, is the coefficient of x^100. \n The fo
llowing list has the factor and the multiplicity.\nThis function is us
eful when you want to grab a particular factor, just index the list." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "ffac[2];\nffac[2,1];\nffac[2,1,1];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7+7$,&%\"xG\"\"\"F'!\"\"F'7$,,F'F'F&F(*$)F&\"\"#F'F'*$)
F&\"\"$F'F(*$)F&\"\"%F'F'F'7$,,F'F'*$)F&\"\"&F'F(*$)F&\"#5F'F'*$)F&\"#
:F'F(*$)F&\"#?F'F'F'7$,&F'F'F+F'F'7$,,F?F'F<F'F9F'F6F'F'F'F'7$,,*$)F&
\"#SF'F'*$)F&\"#IF'F(F?F'F9F(F'F'F'7$,,*$)F&\"\")F'F'*$)F&\"\"'F'F(F1F
'F+F(F'F'F'7$,,F1F'F.F'F+F'F&F'F'F'F'7$,&F&F'F'F'F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$,&%\"xG\"\"\"F&!\"\"F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%\"xG\"\"\"F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 26 "Not elegant, but it works." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
206 "Notice that polynomials are not treated as functions.   They are \+
algebraic objects themselves.  \nWe are really computing in the ring Z
[x].\nIf you want to evaluate a polynomial at some value of x, use sub
s()" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f;\nsubs(x=2, f);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(F&F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 21 "The Gaussian integers" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
85 "The Gaussian integers is the set of numbers a+bi with a, b integer
s and i = sqrt(-1);" }}{PARA 0 "" 0 "" {TEXT -1 66 "The next two lines
 are one way to construct the Gaussian integers." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 11 "f := x^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fG,&\"\"\"F&*$)%\"xG\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "alias(i = RootOf(f));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%\"iG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Now i functions as
 you would expect.  Though you have to use evala() to get i^2=-1." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "i^2;\nevala(i^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"iG\"
\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 21 "evala((1+i)*(2+3*i));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&!\"\"\"\"\"*&\"\"&F%%\"iGF%F%" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 56 "Now you can factor polynomials in the Gaussian integ
ers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "factor(y^4-1,i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$**,&%\"yG!\"\"%\"iG\"\"\"F),&F&F)F(F)F),&F&F)F)F)F),&F&F)F)F'F)F'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "1)   Construct other extensio
ns of the integers by adjoining other polynomials.\n      e.g. Z[sqrt(
2)] , Z[sqrt(-2)]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 34 "2)  Experiment with factorization." }}}}}{MARK "6 0 0" 0 
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