Class 10 Ganit – Polynomials (Bahupad) ke Notes

Polynomial kya hota hai?

Polynomial ek algebraic expression hota hai jisme variable, constant aur exponents (whole numbers) hote hain. Jaise: p(x) = 3x² + 2x – 5. Polynomial ko 'bahupad' bhi kehte hain.

Important terms:

• Degree – highest power of variable
• Coefficient – number jo variable ke saath multiply hota hai
• Zeroes – woh values jo polynomial ko zero karein
• Graph – polynomial ka shape (linear, quadratic, cubic)

Is chapter mein hum linear, quadratic aur cubic polynomials ke zeroes nikalna, factor theorem aur graph analysis seekhenge.

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

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सांख्यिकी

प्रायिकता

1. Polynomial ki Degree aur Types

Polynomial ki degree usme present highest exponent hoti hai. Degree ke hisaab se types:

• Linear polynomial (degree 1) – jaise ax + b, zeroes: ek
• Quadratic polynomial (degree 2) – jaise ax² + bx + c, zeroes: do
• Cubic polynomial (degree 3) – jaise ax³ + bx² + cx + d, zeroes: teen

2. Zeroes of Polynomial – Meaning aur Relation with Coefficients

Kisi polynomial p(x) ke liye, x = a ek zero tab hota hai jab p(a) = 0. Example: p(x) = x² – 5x + 6, zeroes 2 aur 3 hain kyunki p(2)=0 aur p(3)=0.

Quadratic polynomial ke liye:
Sum of zeroes = – (coefficient of x) / (coefficient of x²) = –b/a
Product of zeroes = constant term / coefficient of x² = c/a

Cubic polynomial ke liye:
Sum of zeroes = –b/a
Sum of product of zeroes taken two at a time = c/a
Product of zeroes = –d/a

3. Factor Theorem aur Remainder Theorem

Remainder Theorem: Jab polynomial p(x) ko (x – a) se divide karte hain, toh remainder p(a) hota hai.
Factor Theorem: Agar p(a)=0 hai, toh (x – a) polynomial ka factor hota hai. Is theorem ka use polynomials ko factor karne mein hota hai.

Example: p(x)=x³ – 6x² + 11x – 6, check karein kya (x – 1) factor hai? p(1)=1-6+11-6=0, toh (x-1) factor hai. Is tarah hum baaki factors nikal sakte hain.

4. Polynomial ka Graph aur Shape

Linear polynomial ka graph ek straight line hota hai. Quadratic polynomial ka graph parabola (U-shape) hota hai. Agar a>0 toh parabola upward open hota hai, agar a<0 toh downward. Cubic polynomial ka graph S-shaped curve hota hai jo x-axis ko ek, do ya teen baar cross karta hai – number of real zeroes uske according hote hain.

5. Important Problems aur Short Tricks

• Quadratic polynomial banane ke liye jab zeroes given ho: p(x)=k[x² – (sum)x + (product)]
• Division algorithm: p(x)=g(x)×q(x)+r(x) – always use karein while dividing polynomials
• Zeroes nikalne ke liye splitting middle term method ya quadratic formula use karein

Yeh notes aapko Class 10 board exams ke liye madad karenge. Polynomials ke concepts ko ache se samajhne ke liye sabhi examples NCERT book se solve karein.