Lleqcia 12 mravalwevrTa (polinomTa) Teoria moqmedebebi polinomebze, polinomTa gayofadoba, naSTiT gayofis algoriTmi

Lleqcia 12

mravalwevrTa (polinomTa) Teoria

1. moqmedebebi polinomebze, polinomTa gayofadoba, naSTiT gayofis algoriTmi,

2. 
   polinomTa udidesi saerTo gamyofi, misi povnis evklides algoriTmi, Tanamartivi polinomebi,
   
3. 
   dayvanadi da dauyvanadi polinomebi, polinomis daSla mamravlebad, jeradi dauyvanadi mamravlebi. polinomis fesvi, bezus Teorema
   

4. ricxviT koeficientebiani polinomebi

moqmedebebi polinomebze, polinomTa gayofadoba, naSTiT gayofis algoriTmi,

K veils mimarT n xarisxis f(x) polinoms aqvs Semdegi saxe

f(x)=a₀xⁿ+a₁x^n-1+…+a_n-1x+a_n,

sadac a₀ , a₁, …, a_n elementebia K velidan.

a₀, a₁, …, a_n, elementebs ewodeba f(x) polinomis koeficientebi; a₀xⁿ, a₁x^n-1, …, a_n-1x, a_n-s ewodebaT f(x) polinomis wevrebi. Tu a₀0, maSin a₀-s ewodeba polinomis ufrosi koeficienti, a₀xⁿ-s _ ufrosi wevri, xolo n0 ricxvs _ f(x) polinomis xarisxi da iwereba

n=deg f(x).

Tu n=0, f(x)=a_n≠0, maSin f(x) nulxarisxis polinomia. xolo f(x) polinoms ewodeba nulovani polinomi da iwereba f(x)=0, Tu misi yvela koeficienti nulia e. i. nulovani polinomi aris K velis nulovani elementi; nulovan polinoms xarisxi ar aqvs.

polinomTa simravleSi SemoviRoT Sekrebisa da gamravlebis operaciebi Semdegnairad:

vTqvaT mocemulia ori

f(x)=a₀xⁿ+a₁x^n-1+…+a_n-1x+a_n=_,

da

g(x)=b₀x^m+b₁x^m-1+…+b_m-1x+b_m=_,

polinomi.

maTi jamia f(x)+ g(x)= ,

sadac, Tu , maSin , xolo Tu , maSin an , romelic arsebobs (anu f(x)+ g(x)-is -is koeficienti aris f(x)-is -is koeficientis -isa da g(x)-is -is koeficientis -is jami).

xolo am polinomebis namravli

f(x) g(x)= ,

sadac (j=0, 1, 2, …, ).

kerZod,

d₀=a₀b₀, d₁= a₀b₁+a₁b₀,, d₂= a₀b₂+a₁b₁+ a₂b₀, …,

d_n+m-1=a_n-1b_m+ a_nb_m-1, d_n+m= a_nb_m.

aseTnairad Semotanili Sekrebisa da gamravlebis operaciis mimarT polinomTa simravle komutaciuri rgolia.

vTqvaT f(x), g(x) polinomebia, amasTanave g(x)0.

gansazRvreba. vityviT, rom g(x) yofs f(x)-s, an f(x) iyo­fa g(x)-ze da davwerT g(x)|f(x), Tu ki moiZebneba iseTi q(x) polinomi, rom f(x)=g(x)q(x).

Tu ki aseTi polinomi ar moiZebneba, maSin vityviT, rom g(x) ar yofs f(x)-s da davwerT g(x)†f(x).

Teorema. (naSTiT gayofis algoriTmi) nebismieri f(x) da g(x) polinomebisaTvis, sadac g(x)0, arsebobs q(x) da r(x) polinomTa erTaderTi wyvili iseTi, rom Sesruldeba toloba:

f(x)=g(x)q(x)+r(x), sadac r(x)=0 an deg r(x)x).

q(x)-s ewodeba ganayofi, xolo r(x)-s naSTi.

Tu r(x)=0, maSin g(x) yofs f(x)-s. g(x) aris f(x)-is gamyofi, xolo f(x) aris g(x)-is jeradi.

polinomTa udidesi saerTo gamyofi, misi povnis evklides algoriTmi, Tanamartivi polinomebi,
gansazRvreba vityviT, rom d(x)0 polinomi aris f(x) da g(x) polinomebis saerTo gamyofi, Tu d(x)|f(x) da d(x)|g(x). xolo f(x) da g(x) polinomebis udidesi saerTo gamyofi ewodeba am polinomebis im D(x) saerTo gamyofs, romelic TviTon iyofa maT nebismier saerTo gamyofze. imis aRsaniSnavad, rom D(x) aris f(x) da g(x) polinomebis udidesi saerTo gamyofi weren:

D(x)=(f(x),g(x)).

evklides algoriTmi. vaCvenoT axla, rom yovel or polinoms, romelic erTdroulad nulovani araa. gaaCnia udidesi saerTo gamyofi.

vTqvaT f₁(x) da f₂(x) ori iseTi polinomia, rom f₂(x)0. zogadobis daurRvevlad vigulisxmoT, rom f₂(x)†f₁(x) (radgan Tu f₂(x)|f₁(x) maSin (f₁(x),f₂(x))=f₂(x) ), amitom naSTiT gayofis algoriTmis Tanaxmad, gveqneba: f₁(x)=f₂(x)q₁(x)+f₃(x), sadac f₃(x)=0, an degf₃(x)2(x). Tu f₃(x)=0, maSin process SevwyvetT, xolo Tu degf₃(x)2(x). maSin kvlav naSTiT gayofis algoriTmis ZaliT, f₂(x)=f₃(x)q₂(x)+f₄(x), sadac an f₄(x)=0, an degf₄(x)3(x). Tu f₄(x)=0, maSin process SevwyvetT, xolo Tu degf₄(x)3(x), maSin process ganvagrZobT da a. S. ramdenime nabijis Semdeg, es procesi aucileblad Sewydeba, radgan degf₂(x)>degf₃(x)>degf₄(x)>…

amrigad, gveqneba:

f₁(x)=f₂(x)q₁(x)+f₃(x), degf₃(x)2(x),

f₂(x)=f₃(x)q₂(x)+f₄(x), degf₄(x)3(x),

. . . . . . . . . . . . . . . .

f_i(x)=f_i+1(x)q_i(x)+f_i+2(x), degf_i+2(x)i+1(x),

. . . . . . . . . . . . . . . .

f _k-2(x)=f _k-1(x)q _k-2(x)+f _k(x), degf _k(x)k-1(x),

f _k-1(x)=f _k(x)q _k-1(x).

ukanaskneli tolobidan gamomdinareobs, rom f _k(x)|f _k-1(x); amitom Tu ganvixilavT tolobebs qvemodan zemoT, miviRebT, rom f_k(x)|f₂(x); da f_k(x)|f₁(x); ); e.i. f_k(x) aris f₁(x) da f₂(x) polinomebis saerTo gamyofi. vTqvaT axla, d(x) aris f₁(x) da f₂(x) polinomebis nebis­mi­eri saerTo gamyofi, maSin pirveli tolobidan d(x)|f₃(x); amitom Tu ganvixilavT tolobebs zemodan qvemoT, miviRebT, rom d(x)|f_k(x). maSasadame, f₁(x) da f₂(x) polinomebis saerTo gamyofi f _k(x) iyofa f₁(x) da f₂(x) polinomebis nebismier saerTo gamyofze, e.i. f_k(x)= (f₁(x),f₂(x)).

wess, romlis saSualebiTac vipoveT f₁(x) da f₂(x) polinomebis udidesi saerTo gamyofi (f₁(x),f₂(x)), evklides algoriTmi ewodeba. maSasadame, ori polinomis udi­desi saerTo gamyofi aris evklides algoriTmis ukanaskneli aranulovani naSTi.

gansazRvreba. Tu or f(x) da g(x) polinoms ar gaaCnia saerTo gamyofi garda K velis aranulovani elementebisa, maSin maT Tanamartivi polinomebi ewodeba.


dayvanadi da dauyvanadi polinomebi, polinomis daSla mamravlebad, jeradi dauyvanadi mamravlebi, polinomis fesvi, bezus Teorema
gansazRvreba. vTqvaT f(x) polinomia koeficientebiT K velidan. vityviT, rom f(x) dayvanadia K velis mimarT, Tu moiZe­bneba iseTi ori f₁(x) da f₂(x) polinomi koeficientebiT K velidan, rom f(x)=f₁(x)f₂(x), amasTanave deg f₁(x)₂(x) Tu ki aseTi ori polinomi koeficientebiT K velidan ar moiZebneba, maSin vityviT, rom f(x) dauyvanadia K velis mimarT. ase magaliTad, x²-1 polinomi dayvanadia Q-racionalur ricxvTa velis mimarT, radganac x²-1=(x-1)(x+1); x²-5 polinomi ki dauyvanadia Q-s mimarT, magram dayvanadia R-namdvil ricxvTa velis mimarT, radgan ; x²+1 polinomi dauyvanadia, rogorc Q-s, ise R-is mimarT, magram dayvanadia C-kompleqsur ricxvTa velis mimarT, radgan x²+1=(x-i)(x+i).

samarTliania Semdegi Teorema

Teorema. yoveli f(x) polinomi, romlis xarisxi n>0, K velis mimarT iSleba dauyvanad polinomTa namravlad da es daSla erTaderTia nulxarisxis polinomebamde sizustiT

f(x)=ap₁(x)p₂(x)…p_r(x).

am daSlaSi Semavali dauyvanadi mamravlebi SeiZleba ar iyos gansxvavebuli. Tu romelime dauyvanadi p(x) mamravli daSlaSi gvxvdeba zustad k-jer, e. i.

p^k(x)f(x), magram p^k+1(x)†f(x),

maSin mas f(x) polinomis k-jeradi dauyvanadi mamravli ewodeba. kerZod, Tu p(x) mamravli daSlaSi mxolod erTxel Sedis, e. i. p(x)f(x), magram p²(x) †f(x), maSin mas f(x) polinomis martivi dauyvanadi mamravli ewodeba.

Tu vigulisxmebT, rom daSlaSi p₁(x),…,p_s(x) (1sr) mamravlebi erTmaneTisgan gansxvavebulia da jeradoba Sesabamisad aris k₁,…,k_s, maSin miviRebT daSlas:

,

romelsac ewodeba f(x) polinomis kanonikuri daSla dauyvanad mamravlebad K velis mimarT.

gansazRvreba. c eleme­nts ewodeba f(x) polinomis fesvi Tu f(c)=0, e.i. f(x) po­linomis mniSvneloba, roca x=c, aris nuli.

bezus Teorema. imisaTvis rom c elementi iyos f(x) polinomis fesvi, aucilebeli da sakmarisia, rom x-cf(x).

damtkiceba. sakmarisoba. Tu x-cf(x), maSin f(x)=(x-c)q(x), saidanac

f(c)=(c-c)q(c)=0.

aucilebloba. vTqvaT axla f(c)=0. naSTiT gayofis algoriTmis Tanaxmad, f(x)=(x-c)q(x)+r(x), sadac r(x)=0 an degr(x)<1, e.i.

f(x)=(x-c)q(x)+r, (20)

Tu miRebul tolobaSi dav­werT x-is nacvlad c-s, gveqneba:

f(c)=(c-c)q(c)+r=r,

e.i. r=0 da x-cf(x).

ricxviT koeficientebiani polinomebi
ricxviT koeficientebiani polinomebisaTvis samarTliania Semdegi Teoremebi:
1. kenti xarisxis namdvilkoeficientebian polinoms aqvs erTi mainc namdvili fesvi.

2. 
   namdvilkoeficientebian polinoms, romlis xarisxi n≥1 aqvs erTi mainc kompleqsuri fesvi.
   

2. 
   algebris ZiriTadi Teorema: kompleqsur ko­e­fi­ci­e­ntebian polinoms, romlis xarisxi n1 aqvs erTi mainc komple­qsuri fesvi.
   

4. nebismieri ricxviT koeficientebiani polinomi, romlis xarisxi n1, kompleqsur ricxvTa velis mimarT iSleba dauyvanad polinomTa namravlad. kompleqsur ricxvTa velis mimarT dauyvanadia mxolod pirveli xarisxis polinomebi.