Polynomials in One Variable

🎥 Video Explanation

🇮🇳 Hindi Explanation

🇬🇧 English Explanation

📊 Infographic Summary

Infographic Summary

1. Polynomials in One Variable Kya Hote Hain?

Pichle lesson mein humne dekha ki Polynomials kya hote hain. Ab hum specifically One Variable wale polynomials ki baat karenge.

Simple words mein: Aisa polynomial jisme sirf ek hi type ka letter (variable) use ho.

✔ Examples:

  • 2x² + 3x - 5 (Sirf 'x' hai → One Variable)
  • y³ - y + 1 (Sirf 'y' hai → One Variable)
  • t² + 4 (Sirf 't' hai → One Variable)

📌 Notation: Hum polynomials ko denote karne ke liye p(x), q(x), r(y) etc. use karte hain.

  • p(x) = 2x² + 5x - 3
  • q(y) = y³ + y + 1

❌ Not One Variable:

  • x² + y² (Do variables 'x' aur 'y' hain)
  • a² + b + c (Teen variables 'a', 'b', 'c' hain)

2. Degree of a Polynomial 🎓

📖 Concept: The Highest Power Wins

Imagine karo ek class mein students ke marks aaye hain. Topper kaun hota hai? Jiske marks sabse zyada hote hain. Bilkul waise hi, ek polynomial mein variable ki Highest Power (sabse badi power) ko us polynomial ki Degree kehte hain.

✔ Examples:

  • 5x³ - 4x² + x - √2
    Powers hain: 3, 2, 1. Highest is 3. So, Degree = 3.
  • 2 - y² - y³ + 2y⁸
    Powers hain: 2, 3, 8. Highest is 8. So, Degree = 8.
  • 2 (Constant Polynomial)
    Isse 2x⁰ likh sakte hain. So, Degree = 0.

📌 Note: Zero Polynomial (0) ki degree Not Defined hoti hai.

3. Classification based on Number of Terms 🔢

Polynomials ko hum unke terms ki counting ke hisaab se naam dete hain.

🏷️ Story Time: The Nickname System

Socho aapke friends ka group hai. Agar koi akela hai toh use "Solo" (Monomial) bolte hain. Agar do log hain toh "Duo" (Binomial). Aur agar teen hain toh "Trio" (Trinomial). Maths mein bhi terms count karke hum polynomials ko nickname dete hain!

1️⃣ Monomial

Mono = One

Sirf 1 Term

2x, 5, -3x²
2️⃣ Binomial

Bi = Two

Sirf 2 Terms

x + 1, x² - 5
3️⃣ Trinomial

Tri = Three

Sirf 3 Terms

x² + x + 1

4. Classification based on Degree 📈

Polynomials ko unki degree (highest power) ke hisaab se bhi classify kiya jaata hai.

⚡ Story Time: The Power Levels

Video games mein levels hote hain na? Level 1 (Linear), Level 2 (Quadratic), aur Level 3 (Cubic). Waise hi polynomial ki degree uska "Power Level" batati hai!

📏 Linear

Degree = 1

Straight Line Graph

2x - 1
🔲 Quadratic

Degree = 2

Parabola Graph

x² + 2x + 1
🧊 Cubic

Degree = 3

S-shaped Graph

x³ - 2x

💡 Standard Forms:

  • Linear: ax + b
  • Quadratic: ax² + bx + c
  • Cubic: ax³ + bx² + cx + d

📌 Summary Table

Type Degree Example
Constant 0 7
Linear 1 x + 5
Quadratic 2 2x² + 5
Cubic 3 4x³ - 2x² + 1

📝 Exercise Questions

Chalo check karte hain ki aapko kitna samajh aaya! Neeche diye gaye questions ko solve karein.

Q1: Which of the following are polynomials in one variable?

  1. 4x² - 3x + 7
  2. y² + √2
  3. 3√t + t√2
  4. y + 2/y
Show Answer
  1. ✅ Yes (Only 'x', whole number powers)
  2. ✅ Yes (Only 'y', whole number powers)
  3. ❌ No (Power of t is 1/2, not a whole number)
  4. ❌ No (2/y means 2y⁻¹, power is -1)

Q2: Write the degree of each of the following polynomials:

  1. 5x³ + 4x² + 7x
  2. 4 - y²
  3. 5t - √7
  4. 3
Show Answer
  1. Degree = 3 (Highest power is 3)
  2. Degree = 2 (Highest power is 2)
  3. Degree = 1 (Power of t is 1)
  4. Degree = 0 (Constant polynomial)

Q3: Classify the following as Linear, Quadratic, and Cubic polynomials:

  1. x² + x
  2. x - x³
  3. y + y² + 4
  4. 1 + x
  5. 3t
  6. r²
  7. 7x³
Show Answer
  1. Quadratic (Degree 2)
  2. Cubic (Degree 3)
  3. Quadratic (Degree 2)
  4. Linear (Degree 1)
  5. Linear (Degree 1)
  6. Quadratic (Degree 2)
  7. Cubic (Degree 3)

🚀 Next Steps

Great! Ab aapko polynomials ke types samajh aa gaye hain. Next lesson mein hum seekhenge ki Zeroes of a Polynomial kaise nikaalte hain.

Next Topic: Zeroes of a Polynomial →