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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 37 "The result
ant, summary of properties." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 169 "Suppose you have two polynomials f and g in x.
 Say one is of degree 2 and one is of degree 3. Suppose the highest co
efficient of both polynomials is 1, so we can write: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=add(a.i*x^i,i=0..1)+x^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*&%#a1G\"\"\"%\"xGF(F(%#a0GF(*
$)F)\"\"#\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g:=add
(b.i*x^i,i=0..2)+x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,*%#b0G
\"\"\"*&%#b1GF'%\"xGF'F'*&%#b2GF')F*\"\"#\"\"\"F'*$)F*\"\"$F/F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Now lets consider the following ma
trix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M:=sylvester(f,g,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7'7'\"\"\"%#a1G%#a0G
\"\"!F-7'F-F*F+F,F-7'F-F-F*F+F,7'F*%#b2G%#b1G%#b0GF-7'F-F*F1F2F3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d:=det(M);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"dG,<*$)%#b0G\"\"#\"\"\"\"\"\"*(%#a1GF+%#b1GF+F(F
+!\"\"*&%#a0GF+)F.F)F*F+*(F1F*%#b2GF+F(F*!\"#*(F4F*)F-F)F*F(F*F+**F-F*
F.F*F1F*F4F*F/*&)F1F)F*)F4F)F*F+*(F1F*F-F*F(F*\"\"$*&F:F*F.F*F5*&)F-F=
F*F(F*F/*(F7F*F.F*F1F*F+*(F:F*F4F*F-F*F/*$)F1F=F*F+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 77 "The matrix formed in this way from the coeffici
ents of f and g is called the " }{TEXT 259 16 "Sylvester matrix" }
{TEXT -1 48 ". The determinant of that matrix is called the: " }{TEXT 
260 10 "resultant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "R:=re
sultant(f,g,x);  # same as d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG
,<*$)%#b0G\"\"#\"\"\"\"\"\"*(%#a1GF+%#b1GF+F(F+!\"\"*&%#a0GF+)F.F)F*F+
*(F1F*%#b2GF+F(F*!\"#*(F4F*)F-F)F*F(F*F+**F-F*F.F*F1F*F4F*F/*&)F1F)F*)
F4F)F*F+*(F1F*F-F*F(F*\"\"$*&F:F*F.F*F5*&)F-F=F*F(F*F/*(F7F*F.F*F1F*F+
*(F:F*F4F*F-F*F/*$)F1F=F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "W
e claim that the resultant equals zero if and only if f and g have a r
oot in common. In other words" }}{PARA 0 "" 0 "" {TEXT -1 63 " resulta
nt(f,g,x) = 0   if and only if gcd(f,g) is not trivial." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 180 "Lets check this in an example where we c
an easily see when f,g have a common root, namely in case that we know
 the roots alpha.i of f and beta.j of g. Then we can write f and g as:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=mul(x-alpha.i,i=1..2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,&%\"xG\"\"\"%'alpha1G!
\"\"F(,&F'F(%'alpha2GF*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "g:=mul(x-beta.j,j=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG
*(,&%\"xG\"\"\"%&beta1G!\"\"F(,&F'F(%&beta2GF*F(,&F'F(%&beta3GF*F(" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Now f and g have a non-trivial g
cd if and only if there exists an i and a j such that alpha.i=beta.j, \+
which happens if and only if the following product equals 0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P:=mul(mul(alpha.i-beta.j,i=
1..2),j=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*.,&%'alpha1G
\"\"\"%&beta1G!\"\"F(,&%'alpha2GF(F)F*F(,&F'F(%&beta2GF*F(,&F,F(F.F*F(
,&F'F(%&beta3GF*F(,&F,F(F1F*F(" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 
16 "And we see that:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "R:=
resultant(f,g,x);  # is the same." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"RG*.,&%'alpha1G\"\"\"%&beta1G!\"\"F(,&%'alpha2GF(F)F*F(,&F'F(%&beta
2GF*F(,&F,F(F.F*F(,&F'F(%&beta3GF*F(,&F,F(F1F*F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 54 "We see that the polynomial P equals the resultant.
 So:" }}{PARA 0 "" 0 "" {TEXT 258 93 "resultant(f,g,x) is an expressio
n which is zero if and only if f and g have a root in common." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 8 "Theorem:
" }}{PARA 0 "" 0 "" {TEXT -1 43 "Let f = the product of x-alpha.i for \+
i=1..n" }}{PARA 0 "" 0 "" {TEXT -1 41 "Let g= the product of x-beta.j \+
for j=1..m" }}{PARA 0 "" 0 "" {TEXT -1 123 "Then resultant(f,g,x), whi
ch is defined as det(sylvester(f,g,x)), equals the product of all alph
a.i-beta.j, i=1..n, j=1..m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 58 "Here is an example where the leading coefficien
t is not 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=3*x^5+2*x
^4+9*x^2+a*x-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*$)%\"xG\"
\"&\"\"\"\"\"$*$)F(\"\"%F*\"\"#*$)F(F/F*\"\"**&%\"aG\"\"\"F(F5F5!\"$F5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=-3*x^2+5*x-8;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*$)%\"xG\"\"#\"\"\"!\"$F(\"\"&
!\")\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sylvester(f,g
,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)\"\"$\"\"#\"\"
!\"\"*%\"aG!\"$F*7)F*F(F)F*F+F,F-7)F-\"\"&!\")F*F*F*F*7)F*F-F0F1F*F*F*
7)F*F*F-F0F1F*F*7)F*F*F*F-F0F1F*7)F*F*F*F*F-F0F1" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "d:=det(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"dG,(!'(yG\"\"\"\"%\"aG\"&Fx\"*$)F(\"\"#\"\"\"!$['" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(d);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(!'(yG\"\"\"\"%\"aG\"&Fx\"*$)F&\"\"#\"\"\"!$['" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a1,a2:=solve(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>6$%#a1G%#a2G6$,&#\"%4f\"$K%\"\"\"*&%\"IGF,-
%%sqrtG6#\"#r\"\"\"#\"$v\"F+,&F)F,F-#!$v\"F+" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "gcd(f,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "op(subs(a=9,[f,g]));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*$)%\"xG\"\"&\"\"\"\"\"$*$)F&\"
\"%F(\"\"#*$)F&F-F(\"\"*F&F0!\"$\"\"\",(F.F1F&F'!\")F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "op
(subs(a=a1,[f,g]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*$)%\"xG\"\"&
\"\"\"\"\"$*$)F&\"\"%F(\"\"#*$)F&F-F(\"\"**&,&#\"%4f\"$K%\"\"\"*&%\"IG
F6-%%sqrtG6#\"#rF(#\"$v\"F5F6F&F6F6!\"$F6,(F.F?F&F'!\")F6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"xG\"\"\"#!\"&\"\"'F%*&%\"IGF%-%%sqrtG6#\"#r\"\"\"#
!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(f,g);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 129 "We see that only if we substitute a number for a such th
at the resultant becomes zero, only then will the gcd become non-trivi
al." }}}}{MARK "0 0 1" 37 }{VIEWOPTS 1 1 0 1 1 1803 }