Hello friend!! Aaj hum padhenge Class-9 ka ek very important chapter jiska naam hai, "polynomial". Polynomial ko hindi me bahupad (बहुपद) kaha jata hai. Bahupad math ka bahut jaada important chapter hai. Jisko aaj hum ekdam basic se aur full details ke saath padhenge.
Aaj hum jo kuch bhi padhenge aur sikhenge usse aap nichey diye Table of content me dekh sakte hai.
Table of content
- Beejeey vyanjak (algebraic expression kya hai?
- Char raashi (variable) kya hai?
- Achar raashi (constant) kya hai?
- Bahupad (polynomial) kya hota hai? Yaa Bahupad kise kahte hai?
- Bahupad ka vyapak rup (General form of polynomial)
- Ek char waala bahupad (polynomials in one variable)
- Achar bahupad (constant polynomial)
- Zero polynomial
- Bahupad ke pad (terms of polynomial)
- Bahupad ke pad ka gurnank (coefficient of polynomial's terms)
- Bahupad ki ghaat (degree of the polynomial)
- Bahupad ka vargikaran (classification of polynomials)
- Polynomial ko kaise pahchane? (How to identify a polynomial?)
Upar diye inhi topic ko hum full details me padhane jaa rahe hai. Toh chaliye start karte hai.
Polynomial (bahupad) ka Basic concept, Definition, Examples, Types
Bahupad ko padhane se pahle kuch important points hai jinko hume pahle samjhna hoga. Tabhi hume polynomial acche se samjh aayega. Jaise hume sabse pahle jaana hoga ki, Beejeey vyanjak kya hai?, Char rashi kya hoti hai?, Achar rashi kisko kahte hai? inko hum sabse pahle jaanenge uske baad hum polynomial ko padhenge. Toh chaliye pahle hum inko jaan lete hai.
Beejeey vyanjak (algebraic expression) kya hai?
Sabse pahle hum baat karenge, Beejeey vyanjak ke baare me, Beejeey vyanjak kya hai?
Beejeey vyanjak (algebraic expression) ko samjhne ke liye aap inn examples ko dekhiye -:
2x, 2y+1, 7x+2y-4, x/2-7z
inn sabhi examples ko hum "Beejeey vyanjak" kahte hai. Ek Beejeey vyanjak me kam se kam ek Char raashi aur ek Achar raashi jarur hota hai. Beejeey vyanjak ko hum iss prakar se paribhasit (define) kar sakte hai ki, "char raashi aur achar raashi ka gunanphal (multiplication) beejeey vyanjak hota hai."
Means, Beejeey vyanjak = (Char)×(Achar)
Jaise-: char = y aur achar = 5
Tab, Beejeey vyanjak = 5 × y = 5y
Issi prakar se 2y, 3t-1, 3x+2y+1, z/2 ye sabhi beejeey vyanjak hai. Inn sabhi beejeey vyanjako me jo x, y, z, aur t hai inko char raashi kaha jaata hai aur jo numbers hai 1, 2, 3 aur 5 hai inko achar raashi kaha jaata hai.
Aayiye Char aur Achar rashiyon ke baare me thoda aur acche se jaan lete hai.
Char (चर) variable kya hai ?
Char (चर) variable : Char ko kisi beejeey vyanjak me hum kisi bhi English alphabet jaise a, b, c, d, .... z tak yaa A, B, C, D, ..... Z yaa kisi symbol (prateek yaa chinnh) jaise star (*), hashtag (#), dollar ($) etc. se prakat karte hai.
Jaise:- Beejeey vyanjak 5x+2y me "x" aur "y" char rashi hai.
Dyaan rakhe ki, Char ka koi bhi number value (maan) ho sakta hai. Char ka maan badalta rahta hai.
Jaise-: x+2 = 5 me char x = 3 hoga. Jabki x+4 = 6 me char x = 2 hai. Issiliye hum kah sakte hai ki char raashi ka maan (value) badalta (change) hota rahta hai.
Achar (अचर) constant kya hai ?
Achar (अचर) constant : Number ko hum achar (constant) kahte hai. Jaise-: 1, 2, 3, 4, 5,. . . ye sabhi number Achar (constant) hai.
Kisi Beejeey vyanjak me bhi achar ek fix number hi hota hai jiska maan nahi badalta hai.
Jaise-: Beejeey vyanjak 2x+4 me 2 aur 4 Achar hai. Inka maan humesha 2 aur 4 hi rahega. Ye change nhi honge. Isliye inko hum achar kahenge.
Bahupad (polynomial) kya hota hai? yaa Bahupad kise kahte hai?
Bahupad ko English me Polynomial kaha jata hai. "Polynomial" word, two words "Poly" aur "nominal" se bana hai. Jisme Poly ka meaning bahut (many) aur nominal ka meaning pad (term) hota hai. Means, "Bahupad aisa Beejeey vyanjak hai jisme bahut se pad hote hai."
Ek Bahupad ek ya ek se adhik Beejeey vyanjako (algebraic expression) ke Addition (+) aur Subtraction (-) se milkar banta hai. Inn beejeey vyanjako me char (Variable) ki ghaat (power) whole number hota hai.
Jaise-: 3x+4 ek polynomial hai, jo two beejeey vyanjak 3x aur 4 ke Addition se bana hai. Isme Char x ki ghaat 1 hai, joki whole number hai.
Toh Bahupad ko hum iss prakar se paribhasit kar sakte hai ki, "Wah beejeey vyanjak, jo addition (+) aur subtraction (-) se milkar bane ho aur jisme char ki ghaat (power) ek dhanatmak sankhya (positive integer) yaa purn sankhya (whole number) ho to aise beejeey vyanjak ko hum bahupad kahte hai."
Jaise-: 3x² - 2x + 5, Bahupad ka example hai.
Iss prakar se 2x² - 2x + 5 bhi ek polynomial hai.
Bahupad ka vyaapak roop (General form of polynomial)
Bahupad ke vyaapak roop (general form) ko hum iss prakar se prakat kar sakte hai :-
f(x) = aₒ + a₁x + a₂x² + . . . . anxⁿ
Iss vyaapak roop me aₒ, a₁, a₂, . . . an achar raashi aur x, x², . . . xⁿ char raashi hai.
Ek char waala bahupad (polynomials in one variable)
Ek char waala bahupad, jaisa ki iske naam se pata chal raha hai ki, "Wo bahupad jisme kewal ek char (variable) hote hai, unhe hum "Ek char waala bahupad" kahte hai."
Jaise-: 2x+3 ek char waala bahupad hai kyuki isme kewal ek char x hai.
Issi prakar bahupad x²+2x-3 me ek char x, y²+2 me ek char y, 4z-3z²+7z³ me ek char z hai. Isliye ye sabhi bahupad ek char waale bahupad hai.
Ek char waale polynomial ko hum iss prakar se prakat karte hai-:
p(x) = x²+2x-3
r(y) = y²+2
q(z) = 4z-3z²+7z³
Achar bahupad (constant polynomial)
Achar bahupad aisa bahupad hota hai jisme kewal achar pad hote hai. Jaise -: 2 ek achar bahupad hai, kyuki issme kewal ek achar pad 2 hai.
Issi prakar se 5, -4, 1 etc. achar bahupad ke examples hai.
Zero bahupad (zero polynomial)
Zero "0" bhi ek Achar bahupad hai. Jisko hum Zero bahupad kahte hai.
Bahupad ke pad (terms of polynomial)
Bahupad x²+2x+9 me x², 2x aur 9 ye bahupad ke "pad (terms)" kahlate hai. Toh iss prakar se hum kah sakte hai ki, Bahupad x²+2x+9 me total 3 pad (three terms) hai.
Issi prakar se , Bahupad y+8 me y aur 8 total 2 pad, z²+2z³-5z+7 me z², 2z³, -5z aur 7 total 4 pad hai.
Bahupad ke pad ka gurnank (Coefficient of polynomial's term)
Gurnank ka matalab hota hai, "kisi number se guna (multiply) kiya huaa." Kisi Bahupad ke kisi bhi pad (term) ke aage jo number likha hota hai, Wo number uss pad ka gurnank kahlata hai.
Jaise-: Polynomial 5x²+2x-5 me pad 5x² ke aage 5 hai. Isliye pad 5x² ka gurnank 5 hai. Issi prakar 2x ka gurnank 2 hai.
Isse hum iss prakar se bhi samjh sakte hai, "Bahupad ke kisi pad ke char ka jis number se guna (multiply) kiya huaa hota hai. Uss number ko hum uss pad ka gurnank kahte hai."
Bahupad ke har pad (Each term) ka ek gurnank (coefficient) jarur hota hai.
Jaise -: Polynomial 4x³-5x²+x-5 me pad 4x³ ke char x³ ka multiply 4 ke saath huaa hai isliye 4, pad 4x³ ka gurnank (coefficient) hai.
Bahupad ke pad ka gurnank pata karne ka sabse aashan tarika ye hai ki, Bahupad ke pad ke aage jo number likha hota hai. Wahi number uss Pad ka gurnank (coefficient) hota hai.
Jaise-: 5x+2 Bahupad ke pad 5x me char x ke aage number 5 hai. Yahi number 5, pad 5x ka gurnank (coefficient) hai.
Ek aur example se samjhiye, ek bahupad z²+2z³-5z+7 me total 4 pad z², 2z³, -5z aur 7 hai.
• z² ka coefficient = 1
[Note-: Jis pad me char ke saath koi number nahi hota hai toh waha hum 1 maan lete hai.]
• 2z³ ka coefficient = 2
• 5z ka coefficient = 5
Bahupad ki ghaat (Degree of the polynomial)
"Ek bahupad me char ki adhikatam ghaat (highest power) waale pad ke ghaatank ko uss bahupad ki ghaat yaa degree of the polynomial kahte hai."
Jaise-: (i) 5x⁵ + 2x⁴ - 3x³ -2 ek char "x" waala bahupad hai. Isme hum dekh sakte hai ki,
x ki adhikatam ghaat (highest power) = 5
Isliye iss bahupad ki ghaat = 5
(ii) y⁴ - y³ -5 ek Char "y" waala bahupad hai. Isme hum dekh sakte hai ki,
y ki adhikatam ghaat = 4
Isliye iss bahupad ki ghaat = 4
Bahupad ka vargikaran (classification of polynomials)
Bahupad ko unke pado (terms) aur ghaanto (power) ke aadhar par baata jaata hai.
Pado (terms) ke aadhar par Bahupad nimnlikhit prakar ke hote hai-:
(1). Ekpadiy Bahupad (Monomial) - Jis Bahupad me kewal ek pad (one term) hote hai. Uss Bahupad ko "Ekpadiy Bahupad" kahate hai.
Jaise-: 5x², -6y³, -2s⁴, 7z etc.
(2). Dwipadiy Bahupad (Binomial) - Jis Bahupad me two pad (two terms) hote hai. Uss Bahupad ko "Dwipadiy Bahupad" kahate hai.
Jaise-: -2x+3, x² - 5, 6y³ - 2y, 2s⁴ + 2s etc.
(3). Tripadiy Bahupad (Trinomial) - Jis Bahupad me teen pad (three terms) hote hai. Uss Bahupad ko "Tripadiy Bahupad" kahate hai.
Jaise-: x² + 2x + 3, y² - 5y - 1, 2u-3u²+4 etc.
Ghaanto (power) ke aadhar par Bahupad nimnlikhit prakar ke hote hai-:
(1). Rakhik Bahupad (Linear polynomials) - Jis Bahupad ka ghaat "1" (one) hota hai. Uss bahupad ko "Rakhik bahupad" kahte hai.
Jaise-: 3x+1, u+3 dono bahupad ki ghaat 1 hai. Issliye inn dono Bahupad ko Rakhik bahupad kahenge.
(2). Dwighaat Bahupad (Quadratic polynomials) - Jis Bahupad ka ghaat "2" (two) hota hai. Uss bahupad ko "Dwighaat bahupad" kahte hai.
Jaise-: 3x²+2x+1, y² - 4, u² +2u inn bahupad ki ghaat 2 hai. Issliye inn dono Bahupad ko "Dwighaat bahupad" kahenge.
(3). Trighaat Bahupad (Cubic polynomials) - Jis Bahupad ka ghaat "3" (three) hota hai. Uss bahupad ko "Trighaat bahupad" kahte hai.
Jaise-: x³+2x²-3x+1, y²+2y³-5, u³+7, inn bahupad ki ghaat 3 hai. Issliye inn bahupad ko "Trighaat bahupad" kahenge.
Polynomial ko kaise pahchane ? (How to identify a polynomial?)
Sabhi beejeey vyanjak polynomial nahi hote hai. Isliye hume polynomial ki pahchan karne aana chahiye. Polynomial ki pahchan karne se related Question NCERT Class 10 Math book Exercise 2.1 me pucha gaya hai.
Polynomial ki pahchan karna ekdum aashan hai. Hume kewal Beejeey vyanjak me ye dekhana hai ki, char ki ghaat (power) ek whole number yaa positive integer hona chahiye.
Jaise-: (i). a + 1/a ek beejeey vyanjak hai. Kya yah beejeey vyanjak polynomial hai?
Toh a + 1/a = a + a⁻¹ likh sakte hai.
Isme hum dekh sakte hai ki char a ki ghaat = -1 hai joki whole number yaa positive integer nahi hai. Isliye ye beejeey vyanjak Bahupad nahi hai.
(ii). Kya √t - 3 beejeey vyanjak hai?
√t - 3 = t1/2 - 3 me hum dekh sakte hai ki char t ki ghaat (power) = 1/2 whole number nahi hai. Isliye ye beejeey vyanjak bhi polynomial nahi hai.
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