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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "L-15 MCS 320 Monday 17 Feb
ruary 2003" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 
"#2. 6 Explain the difference between the symbolic and numerical facto
rization of a polynomial in one variable into linear factors." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Answer: the \+
numerical factorization consists in finding all complex roots.  Symbol
ically, we add formal roots, extending the field of rational numbers.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p :=
 x^3 + 2*x - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*$)%\"xG\"
\"$\"\"\"F**&\"\"#F*F(F*F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "r := fsolve(p,x,complex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG6%^$$!+e#))pE#!#5$!+4:rn9!\"*^$F'$\"+4:rn9F,$\"+:
l(R`%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "nf := (x-r[1])*(
x-r[2])*(x-r[3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#nfG*(,&%\"xG\"
\"\"^$$\"+e#))pE#!#5$\"+4:rn9!\"*F(F(,&F'F(^$F*$!+4:rn9F/F(F(,&F'F($\"
+:l(R`%F,!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(
nf) - p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"\"\"!#5F&)%\"xG\"\"
#F&F&*&$F&!\"*F&F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f
actor(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"F(*&
\"\"#F(F&F(F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ali
as(alpha=RootOf(p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q := evala(p/(x-alpha));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,**$)%\"xG\"\"#\"\"\"F**&%&alpha
GF*F(F*F*F)F**$)F,F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"alias(beta=RootOf(q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&alphaG%%b
etaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q1 := evala(q/(x-be
ta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G,(%\"xG\"\"\"%&alphaGF'
%%betaGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sf := (x-alpha
)*(x-beta)*q1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sfG*(,&%\"xG\"\"
\"%&alphaG!\"\"F(,&F'F(%%betaGF*F(,(F'F(F)F(F,F(F(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 11 "expand(sf);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,.*$)%\"xG\"\"$\"\"\"F(*(F&F(%%betaGF(%&alphaGF(!\"\"*&F&F()F*\"
\"#F(F,*&)F+F/F(F&F(F,*&F*F(F1F(F(*&F.F(F+F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "evala(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(
*$)%\"xG\"\"$\"\"\"F(*&\"\"#F(F&F(F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "evalf(alpha); evalf(beta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+:l(R`%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!+e
#))pE#!#5$\"+4:rn9!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "#2. 8 G
ive the Maple command to build the expression..." }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sum(c['i']*product(x['j','
i'],'j'=0..2),'i'=0..17);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,F**&%\"c
G6#\"\"!\"\"\"&%\"xG6$F(F(F)&F+6$F)F(F)&F+6$\"\"#F(F)F)**&F&6#F)F)&F+6
$F(F)F)&F+6$F)F)F)&F+6$F1F)F)F)**&F&6#F1F)&F+6$F(F1F)&F+6$F)F1F)&F+6$F
1F1F)F)**&F&6#\"\"$F)&F+6$F(FGF)&F+6$F)FGF)&F+6$F1FGF)F)**&F&6#\"\"%F)
&F+6$F(FQF)&F+6$F)FQF)&F+6$F1FQF)F)**&F&6#\"\"&F)&F+6$F(FenF)&F+6$F)Fe
nF)&F+6$F1FenF)F)**&F&6#\"\"'F)&F+6$F(F_oF)&F+6$F)F_oF)&F+6$F1F_oF)F)*
*&F&6#\"\"(F)&F+6$F(FioF)&F+6$F)FioF)&F+6$F1FioF)F)**&F&6#\"\")F)&F+6$
F(FcpF)&F+6$F)FcpF)&F+6$F1FcpF)F)**&F&6#\"\"*F)&F+6$F(F]qF)&F+6$F)F]qF
)&F+6$F1F]qF)F)**&F&6#\"#5F)&F+6$F(FgqF)&F+6$F)FgqF)&F+6$F1FgqF)F)**&F
&6#\"#6F)&F+6$F(FarF)&F+6$F)FarF)&F+6$F1FarF)F)**&F&6#\"#7F)&F+6$F(F[s
F)&F+6$F)F[sF)&F+6$F1F[sF)F)**&F&6#\"#8F)&F+6$F(FesF)&F+6$F)FesF)&F+6$
F1FesF)F)**&F&6#\"#9F)&F+6$F(F_tF)&F+6$F)F_tF)&F+6$F1F_tF)F)**&F&6#\"#
:F)&F+6$F(FitF)&F+6$F)FitF)&F+6$F1FitF)F)**&F&6#\"#;F)&F+6$F(FcuF)&F+6
$F)FcuF)&F+6$F1FcuF)F)**&F&6#\"#<F)&F+6$F(F]vF)&F+6$F)F]vF)&F+6$F1F]vF
)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "#1.9 What is the best way \+
to evaluate a dense polynomial?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 33 "p := randpoly(x,degree=15,dense);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"pG,B\"\"&!\"\"*&\"#z\"\"\")%\"xG\"#:F*F*
*&\"#cF*)F,\"#9F*F**&\"#\\F*)F,\"#8F*F**&\"#jF*)F,\"#7F*F**&\"#dF*)F,
\"#6F*F**&\"#fF*)F,\"#5F*F'*&\"#XF*)F,\"\"*F*F**&\"\")F*)F,FGF*F'*&\"#
$*F*)F,\"\"(F*F'*&\"##*F*)F,\"\"'F*F**&\"#VF*)F,F&F*F**&\"#iF*)F,\"\"%
F*F'*&\"#xF*)F,\"\"$F*F**&\"#mF*)F,\"\"#F*F**&\"#aF*F,F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "codegen[cost](p);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,&*&\"#:\"\"\"%*additionsGF&F&*&\"$?\"F&%0multiplica
tionsGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "codegen[cost]
(convert(p,horner));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#:\"\"\"
%*additionsGF&F&*&F%F&%0multiplicationsGF&F&" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 90 "In general we need d additions and d multiplications to
 evaluate a polynomial of degree d." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "#2.7 Give the Maple command to transform
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p :=
 (x+(z^2+1))*(y-(z^2+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG*&,
(%\"xG\"\"\"*$)%\"zG\"\"#F(F(F(F(F(,(%\"yGF(F)!\"\"F(F/F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pu := algsubs(z^2+1=u,p);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#puG*&,&%\"xG\"\"\"%\"uGF(F(,&%\"yGF(F)!\"
\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "epu := expand(pu);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epuG,**&%\"xG\"\"\"%\"yGF(F(*&F
'F(%\"uGF(!\"\"*&F+F(F)F(F(*$)F+\"\"#F(F," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "subs(u=z^2+1,epu);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,**&%\"xG\"\"\"%\"yGF&F&*&F%F&,&*$)%\"zG\"\"#F&F&F&F&F&!\"\"*&F)F&F
'F&F&*$)F)F-F&F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "#2.3(c) " }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p := 2*x^3 + 4*x + 3;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*&\"\"#\"\"\")%\"xG\"\"$F(F(
*&\"\"%F(F*F(F(F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "alia
s(alpha=RootOf(p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q := evala(p/(x-alpha));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,**&\"\"#\"\"\")%\"xGF'F(F(*(F'F
(%&alphaGF(F*F(F(\"\"%F(*&F'F()F,F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Expand(q*(x-alpha)) mod 5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*&\"\"#\"\"\")%\"xG\"\"$F&F&*&\"\"%F&F(F&F&F)F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "To answer (b):" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Expand((alpha+1)^7) mod
 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%&alphaG\"\"#\"\"\"F(F&F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evala((alpha+1)^7) mod \+
5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'RootOfG6#,(*$)%#_ZG\"\"$\"
\"\"F,*&\"\"#F,F*F,F,\"\"%F,F,*$)F$F.F,F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 14 "To answer (a):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "Irreduc(p) mod 5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F
actor(p) mod 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"#\"\"\")%\"
xG\"\"$F&F&*&\"\"%F&F(F&F&F)F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "
#2.5 (c)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "p := x^16 - 3*x^4 + 9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(
*$)%\"xG\"#;\"\"\"F**&\"\"$F*)F(\"\"%F*!\"\"\"\"*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "codegen[C](p,'optimized');" }}{PARA 6 "" 
1 "" {TEXT -1 15 "      t1 = x*x;" }}{PARA 6 "" 1 "" {TEXT -1 17 "    \+
  t2 = t1*t1;" }}{PARA 6 "" 1 "" {TEXT -1 17 "      t3 = t2*t2;" }}
{PARA 6 "" 1 "" {TEXT -1 17 "      t4 = t3*t3;" }}{PARA 6 "" 1 "" 
{TEXT -1 25 "      t6 = t4-3.0*t2+9.0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "py := algsubs(x^4=y,p);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#pyG,(*&\"\"$\"\"\"%\"yGF(!\"\"*$)F)\"\"%F(F(\"\"*F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pyz := algsubs(y^2=z,py);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pyzG,(*&\"\"$\"\"\"%\"yGF(!\"\"\"
\"*F(*$)%\"zG\"\"#F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "For thi
s polynomial, we need 5 multiplications, one addition and one subtract
ion." }{MPLTEXT 1 0 0 "" }}}}{MARK "45 0 0" 81 }{VIEWOPTS 1 1 0 3 2 
1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }