Class 10 Ganit – Polynomials (Bahupad) ke Notes

Polynomial kya hota hai?

Polynomial, yaani bahupad, ek algebraic expression hai—bas itna samajh lo ki isme variable, constant, aur exponents hote hain, aur exponents hamesha whole numbers (0, 1, 2...) hote hain. Example ke liye, p(x) = 3x² + 2x – 5. Isme x variable hai, 3, 2, aur -5 constants hain, aur 2 exponent hai. Aur haan, polynomial ko Hindi mein 'bahupad' kehte hain—donon naam chalte hain.

Important terms:

• Degree? That's just the highest power a variable gets raised to in a term. Simple as that.
• Coefficient—matlab wo number jo variable ke saath juda hota hai, seedha uske saath multiply hota hai. Jaise agar aap ke paas 3x hai, toh 3 hi coefficient hai. Simple si baat hai, bas itna yaad rakho.
• Zeroes — simple enough, they’re the values that make a polynomial go to zero. Plug one in, and the whole thing just collapses to nothing. That’s it.
• Graph – polynomial ka shape, matlab linear, quadratic, ya cubic. Simple.

Is chapter mein hum seekhenge ke linear, quadratic, aur cubic polynomials ke zeroes kaise nikalein. Factor theorem bhi samjhenge, aur graph analysis ki madad se polynomials ko kaise decode karein. Yeh sab kuch step-by-step, theek hai?

Here we have provided NCERT notes for Class 10 गणित in hindi Language, Just select the chapters below to get notes of the same:

वास्तविक संख्याएँ

बहुपद

दो चरों वाले रखिक समीकरणों का युग्म

द्विघात समीकरण

समान्तर श्रेढ़ी

त्रिभुज

निर्देशांक ज्यामिति

त्रिकोणमिति का परिचय

त्रिकोणमिति का अनुप्रयोग

वृत्त

रचनाएँ

वृतों से सम्बंधित क्षेत्रफल

पृष्ठीय क्षेत्रफल एवं आयतन

सांख्यिकी

प्रायिकता

1. Polynomial ki Degree aur Types

Polynomial ki degree hoti hai usmein maujood sabse bada exponent. Bas, yahi ek line kaafi hai samajhne ke liye. Ab degree ke hisaab se types ki baat karein, toh woh alag-alag naam se jaani jaati hain.

• Linear polynomial—that’s degree 1—something like ax + b. And here’s the kicker: it gives you exactly one zero.
• Look at ax² + bx + c — that’s your quadratic polynomial, degree 2. It’s got exactly two zeroes. Always — no exceptions. The curve bends once, comes down, touches the x-axis at those two spots, and moves on. Sometimes they’re real, sometimes they’re complex, but you always get two of them when you count them properly—with multiplicity and all that. That’s just how quadratics roll.
• A cubic polynomial has degree 3, meaning the highest power of x is 3. Something like ax³ + bx² + cx + d. And here's the kicker—it can have up to three zeroes. Teen zeroes, exactly — that's just how it rolls with cubics.

2. Zeroes of Polynomial – Meaning aur Relation with Coefficients

Kisi bhi polynomial p(x) ke liye, x = a ek zero tabhi hota hai jab p(a) = 0. Matlab, us value ko plug karo aur answer zero aana chahiye. Example le lo: p(x) = x² – 5x + 6. Iske zeroes 2 aur 3 hain — kyunki p(2) = 0 aur p(3) = 0, dono jagah equation poora zero ho jaata hai.

Quadratic polynomial ka case lo — wahan zeroes ka sum nikalna ho toh formula hai: – (coefficient of x) / (coefficient of x²). Matlab –b/a — aur product? Wo hota hai constant term divided by coefficient of x², yani c/a. Bas, itna hi yaad rakho.

For a cubic polynomial, things follow a similar pattern, but with an extra twist. The sum of the zeroes comes out to –b/a. Then, you take the zeroes two at a time, multiply them, and add those up—that gives you c/a. And finally, the product of all three zeroes is –d/a. It’s a neat little chain once you see it, even if it looks a bit heavy at first.

3. Factor Theorem aur Remainder Theorem

Jab aap kisi polynomial p(x) ko (x – a) se divide karte hain, toh jo bachta hai wo remainder hota hai—aur wo remainder exactly p(a) ke barabar hota hai. Simple si baat hai. Ab Factor Theorem isi ko thoda aage le jaati hai: agar p(a) zero ban jata hai, matlab (x – a) us polynomial ka factor hai, bilkul pakka. Aur yahi theorem aapke kaam aati hai jab polynomials ko factor karna ho, toh samajh lo ye dono milke aapka kaam aasaan kar dete hain.

Example: p(x) = x³ – 6x² + 11x – 6. Ab isme check karna hai ki kya (x – 1) factor hai ya nahi. Bas x ki jagah 1 rakho — p(1) = 1 – 6 + 11 – 6 = 0. Zero aaya, matlab (x – 1) bilkul factor hai. Aur aise hi chalte jao, baaki ke factors bhi nikal jaate hain, koi jhanjhat nahi.

4. Polynomial ka Graph aur Shape

Linear polynomial ka graph seedha hota hai—ek simple straight line, koi jhanjhat nahi. Quadratic ka parabola banta hai, woh U-shape wali curve. Agar a > 0 hai, toh U upar khula dikhta hai; a < 0 hone par ulta, neeche. Ab cubic ki baat karein, toh uska graph S-shaped curve hota hai, aur yeh x-axis ko kabhi ek baar, kabhi do, kabhi teen baar cross karta hai. Aur guess karo? Number of real zeroes bhi bilkul wahi hote hain—jitni baar cross, utne hi zeroes.

5. Important Problems aur Short Tricks

• Quadratic polynomial banane ka kaam tab bahut easy ho jaata hai jab aapko zeroes already diye hote hain. Sirf ek formula yaad rakho: p(x) = k[x² – (sum of zeroes)x + (product of zeroes)]. Aur haan, woh 'k' koi bhi non-zero constant ho sakta hai—iska matlab hai ki ek nahi, anek polynomials possible hain. Bas sum aur product nikaalo, formula mein daalo, aur kaam khatam. Koi jhanjhat nahi.
• Division algorithm ka basic funda yaad rakho: p(x) = g(x) × q(x) + r(x). Polynomial divide karte waqt is formula ko hamesha apne saamne rakho, warna galti pakki hai. Yehi equation hai jo aapko har step par guide karti hai, chahe sawal easy ho ya tough. Bas isko apna rule bana lo, aur division ke sawaal aasan lagne lagenge.
• Zeroes nikalne ho, toh dono kaam ke methods hain—splitting middle term ya seedha quadratic formula. Jo aapko comfortable lage, wahi use karein. Middle term split karna shortcut hai, lekin kabhi kabhi formula hi zyada safe rehta hai. Dono apni jagah solid hain, bas practice karte raho.

Yeh notes sach mein kaam aayenge Class 10 ke board exams ke liye. Bas ek baat yaad rakhna—polynomials ka concept tabhi pakka hoga jab tum NCERT ke saare examples khud solve karoge. Haan, khud se, pen utha kar. Bina examples chhode, har ek ko. Isi se clear hoga sab kuch.