Mathematical Methods 1 Lecture notes (MAST4006) Chapter 2- Polynomials

DeFintioY' ·.
f\ ro\~l'\OP\ial f in X is QY\ exptessi or,

f• fO • f 1 )C -f t I 'l(
1
·+• •. -' FII "ll "




DeHniVion:

Li~ bt C\ ~~l_jl'Cm ·•"'' ih -x . "Then \-he dc:jr«e a, f is " Where )'"I 1s 1-h,
\cu9ts~ f\Ul"t\'oer SI-\- . . rf\ -/:·0

~'>1~1"1\ ple :

1+ 1t-+ ,t 1 s: f de3(f)-.1
1-+7(~:: j dtj l3)=1


-
Note :


s~rnbo' s
(c,e.f Htim\- .. ()c ree
1 2
1
--+4">l
1
3
2c -+r<,.. .f "=lx
'j
fR i
... h -+ l1+t) ll 1 C 3
~;-2j1?l-l fxS ~olj. in 5

De.Ho i\-ion:
le~ t,~ 'bi ?Ci\jh<>l'l\i~\s '" J'. l1ntl"I ~=-j iH \J \< e z>,o I
H1d~jlk)

~ro\\~r~

le~ f I ~ nt 90\jnOO\iols d·\--. f,5. 'Th{n dt;) l~) = dl!jl~)
Yroob
SU\)?OSC chj l~) :.n , Tucn t" -:tO o.ncl ro, k>n
I
,. :. 0
t°"'
Sin u. f~5, fo, c,.\I ~, ~K"j11.. \icntt 91\'fo C\l'\Q

OeHni\-ion:

Le~ ~.9 bt pol~noll'\iols
f~ .. ")(."




Suppo~e de.jlfl-:n .,, d~-(jL

Thth P+~ =-
"'
LU . . ~3K )·x:" . -

, fo-t~~_,. (f1~9, )?' -t )JC 2 +. . . -+ (fn -+j")xn

di3 (fAs): n Y)'\CI')( (dt.9CC) ;1cle..9l9)) Nc."rlj


tht"
4 '



7'h lO <tiN' ·.
for pol~"omials f ,-9,

dt~ l'Y'~>< l dij{O ,dt~l~)) l 'rrt


H dej (~) -=t cltjl~) , ~he·n '1. t
dc5 l~-1~)-= mo-,. {clt-5(F), d,3l!,))




J 4-:

H 71" rh e"' k -f 3 I( I '; 0;t O: 0
dej U+-5) n fnO)( (d~(n ,de-5L~))

SuppoH . dcj (f) dtjl-5)
Then f n-+ 311 :: .F 11 +t) ..,. F., O
dtj u. j) "" Q11d - ·a'e-9 U-+,5) ti\
=, dt~ U1_5)-:h .


ltt and 3 be pol~Mrnlal.s

f: t
(•0
i-. :X. 11. 1
J
::. 2 Qj X j
j-:OJ

, de.jU)-:2, clt3l-5~:3

f,'.F t~o +~, )t-+ ~,. ~'!. )\~o-4 3.-~-+j,,c. +<:)3')'.. 1 1)




• f,j, • f, 3•" • f,3, .,, • f • 9,,' • ~. 5, • 1,j, -,, + f, 9,-,:, + f' 5, "" • 1,5,x' • f,3 ,~' • f,j ,>"
5
+~i__53X




le\.- 1 be pol~rnorri"1a\J ,
1"'en tit~ l\~): dt5ln +a{3l~)
~~()CJ~:


lct dt5\0=n ahd dt:9l )=1'h
3
1




l°hth t'nc. lm-1-n )t>-1 c,oeffilleM Ci~ fj l()r rhc tot.H-. 0~ x"' .. ") l·S tn.,....,
Since dc,9(f):n, ~c 1,c.no',J f-"$0
dc9 l9)-="', we. . v-no.\J'. : gn- '1<0
so t"3"" *O
~I" , '"J hi f" ord,r to,"' "'"'' b• 0 ,, n on d m o" ti,, 10,9« I
ordu s horo nocl ' 3

ctt~HC)) oefn~ dtjlj)
Exo""p\e:
~:: \-0( +)t "\

.. \-')l-~,.

~\CJ:: 2
de3 lf+~) = o
f1:: (l+~+~L)l\-')(.-)('!.)

:: \-J. 'l
-'X. +X,-')l
,.
-1(_
)
+')(.
).
-')(.
)
-
')(4



: \ .7 1~-i-l.,c.1_-,."




lft h nit-i on =
F0( po I~Y"IOrY'\IC\
· \ f,9.• \.1H SO\\ v ~InJ iH \-hut e.:Xi.s\..1 I · ,
a pc~nom,a C\1 5\-h . 0
-...J =9~

t~o.1'19\c :
'J. \ ~+t~ 51"Cc.