bahupad

7 ghaat vaale ek bahupad ka kaarteeya nireshaank pranaali mein graaf

praarambhik beejaganit mein dhan (+) aur rin (-) chihnon se sanbanddh kai padon ke vyanjak (expression) ko bahupad (Polynomial) kehte hain, yatha (3a+2b-5c) .

padon ki sankhya ke anusaar iske vishisht upanaam 'ekapad' (monomial), 'dvipd' (binomial), aadi hote hain. uchchatar ganit mein bahupad ka vishisht upayog aise vyanjak ke liye hota hai jiske padon mein kisi ek char raashi, ya ek se adhik char raashiyon, ke shoonya athva dhan poornaank ghaat aaroh ya avaroh kram mein ho, yatha

3x + x2 - x4 . . . . . . . . . . . . (1)
-6x6y + 5px2yx2 - a x . . . . . . . . . . . . (2)

vyanjak (1) (x) ka bahupad hai aur (2) x, y z, ka tatha usamein (a) achar (constant) hai. yadi (x) ke sthaan mein sarvatr koi anya vyanjak maan lein, log x rakh diya jaae, to naya vyanjak log x ka vyanjak kahalaaega. padon ke ghaaton mein se mahattam ko bahupad ka ghaat (degree) kehte hain. yadi ek se adhik char raashiyaaain hon, to vibhinn padon mein char raashiyon ke ghaaton ke yogafalon mein se mahattam ko bahupad ka ghaat kehte hain. is prakaar bahupad (1) ka ghaat 4 hai aur (2) ka 7. aisa bhi kaha jaata hai ki bahupad (2) (x) mein chhathe ghaat ka aur (y) mein dviteeya ghaat ka hai.

do bahupadon ka yogafal, antar aur gunanafal bahupad hi hota hai, kintu unka bhaagafal bahupad naheen hota. do bahupadon ke bhaagafal ko, jinmein ek sankhya maatr bhi ho sakta hai, parimeya falan (rational function) kehte hain. char (x) mein ghaat (m) ka vyaapak bahupad yeh hai :

ao xm +a1 xm-1+.....+am, jahaaain ao ashoonya hai

anukram

beejaganit ka ek maulik prameya

beejaganit ka ek maulik prameya yeh hai ki yadi f (x) char raashi x mein ghaat m ka bahupad hai, to bahupad sameekaran f(x) = 0 ke sada m mool hote hain. ye mool sanmishr (complex) bhi ho sakte hain aur sanpaati (coincident) bhi.

yadi f(x) = 0 ka koi mool p1 hai to bahupad f(x) mein (x-p1) ka bhaag poora-poora chala jaata hai aur bhaagafal mein ghaat m-1 wala ek bahupad f1(x) praapt hota hai. ab bahupad sameekaran f1(x) = 0 ke m-1 mool honge aur yadi iska ek mool (x-p2) hai (yeh bhi sambhav hai ki p2=p1), to fir f1(x) mein (x-p2) ka bhaag poora chala jaaega. yeh ek mahatvapoorn prameya hai ki f(x) ka gunanakhandan adviteeya hota hai.

yadi ham f(x) ke gunaankon aur gunanakhandon mein prayukt sankhyaaon par yeh pratibandh laga dein ki ve kisi amuk kshetr ki hongi, to moolon ka astitv avashyambhaavi naheen rahata. itna avashya hai ki yadi bahupad ka gunanakhandan ho sakega, to gunanakhand adviteeya honge.

vibhinn shaakhaaon mein bahupad ka upayog

trikonamiti ka ek mahatvapoorn prameya yeh hai ki yadi m koi dhanaatmak poornaank hai, to kojya mx ki abhivyakti kojya x ke m ghaatavaale bahupad ke roop mein ki ja sakti hai, yatha

kojya 2x = 2 kojya2 x - 1 ;
kojya 3x = 4 kojya3 x - 3 kojya x

jya mx ke baare mein prameya yeh hai ki yadi m visham hai to jya mx ki abhivyakti jya x ke m vein ghaat ke bahupad ke roop mein ki ja sakti hai aur yadi m sam hai to jya mx / kojya x ki abhivyakti jya x ke m-1 vein ghaat ke bahupad ke roop mein hogi, yatha

jya 3 x = 3 jya x - 4 jya3 x,
jya 4 x = 4 kojya x (jya x - 2 jya3 x)

vaishleshik jyomiti mein vakron ka adhyayan unhein do charon ke bahupad sameekaran dvaara niroopit kar kiya jaata hai. isi prakaar talon ke adhyayan ke liye teen charavaale bahupad sameekaranon ki sahaayata li jaati hai. svechh ghaat ke bahupad sameekaranon se niroopit vakron aur talon ka adhyayan beejeeya jyaamiti mein kiya jaata hai.

do ya adhik charon ke aise bahupad ko, jiske pratyek pad mein charon ke ghaaton ka yogafal samaan ho, samaghaat bahupad, ya keval samaghaat, kehte hain; aadhunik beejaganit mein in samaghaaton ke roopaantaran ka aur in roopaatanranon se sambandhit nishchar (invariant) aur sahaparivrt (covariant) ke siddhaaton ka pramukh sthaan hai aur inke anek upayog hain.

kalan mein ek charavaale bahupad saral varg ke falan hain, kyonki inke avakalan tatha samaakalan ke niyam vishesh roop se saral hain aur har sthiti mein fal ek bahupad hota hai. aadhunik falan siddhaant mein pratyek bahupad apne charon ka ek satat aur vaishleshik falan hota hai. is siddhaant mein ek mahatvapoorn prameya yeh hai ki yadi sanmishr char ka koi falan char ke pratyek parimit maan ke liye vaishleshik hai, to vah ek bahupad hi hoga aur yadi char ke aparimit hone par bhi falan parimit rahata he, to vah keval ek achar hai.

anya upayog

bahupadon ka upayog sanniktan ke liye bhi hota hai. praanrabhik vishleshan ke maanak falan, maikalaurin athva Taylor prameya ke anusaar, ghaat shreni dvaara niroopit kiye ja sakte hain. kaarl vaayastrasi ne 1885 E. mein siddh kiya tha ki koi bhi satat falan kisi bhi koti ki yathaarthata tak ek samaan sanniktan ke saath bahupad dvaara niroopit kiya ja sakta hai.

vishisht bahupad

kisi falan ko vyakt karne ke liye ya, ya2, ....ke atirikt anya bahupad samudaaya bhi hain. udaaharanat:, lajaandr bahupad (Legendre Polynomial). in bahupadon ka upayog anuprayukt ganit mein bahulata se hota hai. isi prakaar harmaait bahupadon ka saankhyiki mein upayog hota hai.

antarveshan (intarapoleshan) samoocha hi bahupad dvaara sannikteekaran (approximation) par aadhaarit hai. (m) diye hue maanon ka upayog karanevaale antarveshan sootr ke aadhaar mein in maanon ko grahan karanevaale m-1 ghaat ke bahupad ki kalpana nihit hoti hai.

baahari kadiyaaain