What is polynomial and equation ?

Tuesday, February 06, 2018

Hello friends, How are you doing? I hope you are doing good!

Friends me hoon Dheeraj Sahni and you are on my website www.mathshindi.com right now. Aapka bohot bohot swagat hai meri iss website par. Iss website par aapko maths se related complex topic ko easy way me samjhaaya jaata hai aur sirf ye hi nahi wo complex topics aapko acche se samajh me aa jaaye iske liye aapko unn topics se related examples/questions bhi diye jaate hai.

Friends, aaj me aapko batane waala hoon polynomials aur equations ke baare me. Iss article mein me aapko bataoonga ki algebraic expression, polynomial, equation etc. kya hota hai. So friends...

Lets begin...

Polynomial aur equation kya hote hain? Ye samajhne ke liye chaliye basic level se start karte hain. Ise dhyaan se samajhna:-

1. Constant: Kisi bhi number(real number) ko constant kehte hain kyuki inki value fix/same rehti hai.Example ke liye 3,–6,12.78,1/4,1.333... etc. Hum sabko pata hai ki 3 kitna hota hai. Hum sabko pata hai ki –6 kitna hota hai. Hum sabko pata hai ki 12.78 kitna hota hai. Hum sabko pata hai ki 1/4 kitna hota hai. Hum sabko pata hai ki 1.333... kitna hota hai etc. Inn sabhi numbers ki value fix hai isiliye inhe constant kehte hain.Kabhi kabhi hame pata nahi hota hai ki diye gaye problem/question me constant kya hai.To aisi situation me constant ko English alphabet ke starting ke letters(small letters) se show kar diya jaata hai.Example:- a,b,c,f,g,h etc.

[agar aap nahi jante hain ki real numbers kise kehte hain to ye jaanne ke liye iss link par click kijiye »»» What is complex number,real number,irrational number, rational number,integer,whole number,natural number?].

2. Variable: Jiski value fix nahi hoti hai usse variable kehte hai.Inhe English alphabet ke ending ke letters(small letters) se show kiya jaata hai.Example:- x,y,z,u,v,t etc.

3. Algebraic expression ( ya expression): Constants aur variables ke mixture ko, jisme +, – ,× , ÷(/) bhi include ho, algebraic expression ( ya expression) kehte hain. Example ke liye– 3x+6, –7x–6y+8z , 7.45m–√5n,

x⁵+√x , 5x/(7yz) , ax × by etc. Ye sabhi algebraic expression ( ya expression) hain.

4. Polynomial: Wo algebraic expression jisme saare variable ki power/exponent whole number ho, polynomial kehlaata hai. For example –

3x² + 5yz – 8. Iss algebraic expression me teen(three) variables hain x, y aur z. Isme x ki power 2 hai jo ki ek whole number hai aur y ki power 1 hai ye bhi ek whole number hai aur z ki power bhi 1 hai aur ye bhi ek whole number hai. So algebraic expression 3x² + 5yz – 8 , ko polynomial kahenge.

[Note:- saare polynomial algebraic expression hote hain lekin saare algebraic expression polynomial nahi hote hain].

[agar aap nahi jante hain ki whole numbers kise kehte hain to ye jaanne ke liye iss blue link par click kijiye »»» What is complex number,real number,irrational number, rational number,integer,whole number,natural number?].

Abb polynomial se related kuch cheezein hain jo aapko pata honi hi chahiye:–

(a) Term: Kisi bhi polynomial me jo kuch bhi sirf aur sirf + ya – ke dwaara attach rehta hai usse term kehte hain. Example ke liye- 3x² + 5yz – 8. Iss polynomial me teen(three) terms hain.

First term: 3x²

Second term: 5yz

Third term: –8(agar kisi term ke aage minus ka symbol hai to wo minus ka symbol bhi ussi term me include kiya jaayega).

(b) Constant term: Wo term jisme koi bhi variable naa ho, sirf constant(koi real number) ho, to aise term ko constant term kehte hain. Example ke liye- 3x² + 5yz – 8. Iss polynomial me third term aisa hai jisme koi bhi variable nahi hai. So iss polynomial me iska third term ,yaani –8 , constant term hua.

(c) Term power: Kisi bhi polynomial ke kisi bhi term me jitne bhi variables hote hain unn sabhi variables ko power ko milakar/add karke jo number aata hai usse uss term ki power kehte hain. Example ke liye- 3x² + 5yz – 8. Iss polynomial me teen(three) terms hain.

First term hai 3x². Iss term me ek hi variable hai - x. Aur iss term me x ki power hai 2 . So polynomial 3x² + 5yz – 8 me first term ki power 2 hai.

Second term 5yz hai. Iss term me do(two) variable hain y aur z. y ki power 1 hai aur z ki bhi power 1 hai. To dono ki power ko milakar/add karke hue 2. So polynomial 3x² + 5yz – 8 me second term ki power 2 hai.

Third term hai –8. Iss term me koi bhi variable nahi hai isilye polynomial 3x² + 5yz – 8 me third term ki power 0 hai.

(d) Degree: Kisi polynomial me jiss term ki power highest yaani sabse zyada hoti hai to highest power waali uss term ke power ko uss poore polynomial ki degree kehte hain. Example ke liye- 3x² + 5yz – 8. Iss polynomial me teen(three) terms hain. First term ki power 2 hai. Second term ki power bhi 2 hai. Aur third term ki power 0 hai. Inn teeno powers me se highest power 2 hai. So polynomial 3x² + 5yz – 8 ki degree 2 hui.

(e) Coefficient: Kisi bhi variable ka coefficient wo number ye algebraic expression hota hai jo kisi term me usko chhodne par bachtaa hai.Example ke liye- 3x² + 5yz – 8. Agar aapko poocha jaaye ki iss polynomial me x² ka coefficient kya hai.To aapko sabse pehle ye dekhna hai ki iss polynomial me x² kaun se term me aa raha hai. x² iss polynomial me first term me hai. Ab aapko iss term me se x² ko chhod dena hai. x² ko chhodne par jo uss term me bachega wahi x² ka coefficient hoga. Yaha par first term me se x² ko chhodne par 3 bachtaa hai. So coefficient of x² in the polynomial 3x² + 5yz – 8 is 3. Issi tarah polynomial 3x² + 5yz – 8 me y ka coefficient 5z , z ka coefficient 5y hai.

[Note:- Coefficient hamesha variables ke hi hote hain, constants ke nahi].

5. Types of polynomial: Polynomial ko do(two) basis par divide kiya gaya hai–
(a) On the basis of terms
(b) On the basis of degree

(a) On the basis of terms – Isme sirf ye dekha jaata hai ki kisi polynomial me kitne terms hai aur saath hi saath usme kitne variables hai ye bhi dekha jaata hai.

Agar kisi polynomial me sirf ek term hai aur ek variable ho to uss polynomial ko monomial in one variable kahenge.

Agar kisi polynomial me do(two) term hai aur ek variable ho to uss polynomial ko binomial in one variable kahenge.

Agar kisi polynomial me teen(three) term hai aur ek variable ho to uss polynomial ko trinomial in one variable kahenge.

Agar kisi polynomial me chaar(four) term hai aur ek variable ho to uss polynomial ko tetranomial in one variable kahenge.

Agar kisi polynomial me 5 ya 5 se zyada term hai aur ek variable ho to uss polynomial ko polynomial(ji haa sirf polynomial) in one variable kahenge.

Isi tarah se, agar kisi polynomial me sirf ek term hai aur do(two) variable ho to uss polynomial ko monomial in two variable kahenge.

Agar kisi polynomial me do(two) term hai aur do(two) variable ho to uss polynomial ko binomial in two variable kahenge.

Agar kisi polynomial me teen(three) term hai aur do(two) variable ho to uss polynomial ko trinomial in two variable kahenge.

Agar kisi polynomial me chaar(four) term hai aur do(two) variable ho to uss polynomial ko tetranomial in two variable kahenge.

Agar kisi polynomial me 5 ya 5 se zyada term hai aur do(two) variable ho to uss polynomial ko polynomial in two variable kahenge.

Issi tarah se teen(three) variable waale polynomial ke saath - in three variables aayega, chaar(four) variable waale polynomial ke saath - in four variables aayega, paanch(five) variable waale polynomial ke saath - in five variables aayega and so on.

Shayad aap samajh gye honge ki me kya kehna chah rha hoon.

(b) On the basis of degree – Isme sirf ye dekha jaata hai ki kisi polynomial ki degree kitne hai aur saath hi saath usme kitne variables hai ye bhi dekha jaata hai.

Agar kisi polynomial ki degree ek hai aur usme ek variable ho to uss polynomial ko linear polynomial in one variable kahenge.

Agar kisi polynomial ki degree do(two) hai aur usme ek variable ho to uss polynomial ko quadratic polynomial in one variable kahenge.

Agar kisi polynomial ki degree teen(three) hai aur usme ek variable ho to uss polynomial ko cubic polynomial in one variable kahenge.

Agar kisi polynomial ki degree chaar(four) hai aur usme ek variable ho to uss polynomial ko biquadratic polynomial in one variable kahenge.

Agar kisi polynomial ki degree 5 ya 5 se zyada hai aur usme ek variable ho to uss polynomial ko polynomial(ji haa sirf polynomial) in one variable kahenge.

Isi tarah se, agar kisi polynomial ki degree ek hai aur usme do(two) variable ho to uss polynomial ko linear polynomial in two variable kahenge.

Agar kisi polynomial ki degree do(two) hai aur usme do(two) variable ho to uss polynomial ko quadratic polynomial in two variable kahenge.

Agar kisi polynomial ki degree teen(three) hai aur do(two) variable ho to uss polynomial ko cubic polynomial in two variable kahenge.

Agar kisi polynomial ki degree chaar(four) hai aur do(two) variable ho to uss polynomial ko biquadratic polynomial in two variable kahenge.

Agar kisi polynomial ki degree 5 ya 5 se zyada hai aur do(two) variable ho to uss polynomial ko polynomial in two variable kahenge.

Shayad aap samajh gye honge ki me kya kehna chah rha hoon.

[Note:- Daily life me aur mathematics me bhi, easiness ke liye, polynomial ko poora nahi padha jaata. Example ke iss polynomial ko dekhiye– 4x² + 5x – 9. Agar terms ke basis par dekha jaaye to ye trinomial in one variable hai, lekin generally log ise sirf trinomial kehna pasand karte hain.Aur inke yahi short name zyada popular hain. Isi tarah agar degree ke basis par dekha jaaye to ye quadratic polynomial in one variable hai, but generally isko short me sirf quadratic polynomial kaha jaata hai].

6. Equations: Agar kisi polynomial ko kisi number(real number) ke equal rakh diya jaaye to usse equation kehte hain. Example ke liye– 4x² + 2x – 8=9,

5xy – 7z= –27, –4.5x + 2.8y=9 etc. Degree 1 waale equation ko linear equation, degree 2 waale equation ko quadratic equation, degree 3 waale equation ko cubic equation etc. kehte hain. Example ke liye 4x + 1=7/5 , linear equation hai. –7y² + 5y – 2=0 , ye ek quadratic equation hai. x³ – 7=1 ye ek cubic polynomial hai etc.

I hope ki aapko mera ye article pasand aaya hoga aur mujhe ummeed hai ki aapko algebraic expression, polynomial, equation ke baare me kafi jaankari mili hogi. Maine aapko aasaan sabdo me aapko algebraic expression, polynomial, equation ke baare me samjhaane ki koshish ki hai.
I hope ki aapko sab kuch acche se samajh me aa gaya hoga.Aur agar aapko koi cheez samajh me nahi aayi ho to commemt ke through aap hame bata sakte hain. Agar aapko ye post accha laga ho to comment ke maadhyam se aap hame support kar sakte hai.

Aapne is article ko padhkar kya seekha?

CHECK YOUR KNOWLEDGE
Give answers:
Q.1-Is x+(1/x) a polynomial?
Q.2-Is –8 a polynomial? If yes, then what type of polynomial is this?
Q.3-Can we say 0, a polynomial?
Q.4-What is the degree of polynomial 78 and polynomial 0?
Q.5-How many terms are there in the polynomial 4x ? Categorize it on the basis of terms as well as on the basis of degree.

Aap apne answers hame comment ke through bata sakte hain.

THANKS FOR READING THIS BLOG.

Maths Hindi

Tuesday, February 06, 2018