🔹 1. What is a Polynomial?
Definition
A polynomial is an algebraic expression made up of variables, constants, and powers of variables which are whole numbers.
বহুপদী হ’ল এটা বীজগাণিতিক অভিব্যক্তি য’ত
- সংখ্যা (constants)
- চলক (variables)
- চলকৰ ঘাত (power)
থাকে, আৰু ঘাতবোৰ ঋণাত্মক বা ভগ্নাংশ নহয়।
Examples / উদাহৰণ
- 2x + 3
- x² − 5x + 7
- 4y³
❌ Not polynomials / বহুপদী নহয়
- 1/x (x⁻¹ আছে)
- √x (x¹ᐟ² আছে)
🔹 2. Terms of a Polynomial
Each part separated by + or − is called a term.
প্ৰতিটো + বা − ৰ মাজৰ অংশক term (পদ) কোৱা হয়।
Example:
3x² − 2x + 5
Terms: 3x², −2x, 5
🔹 3. Coefficient
The number multiplied with the variable is called coefficient.
চলকৰ লগত গুণ হোৱা সংখ্যাটোক coefficient কোৱা হয়।
Example:
In 7x² → coefficient = 7
In −3x → coefficient = −3
🔹 4. Constant Term
A term without variable is called constant term.
যি পদত কোনো চলক নাই তাক constant term কোৱা হয়।
Example:
x² + 4x + 9 → Constant = 9
🔹 5. Degree of a Polynomial
The highest power of the variable is called the degree.
চলকৰ সৰ্বোচ্চ ঘাতটোক বহুপদীৰ degree কোৱা হয়।
| Polynomial | Degree |
|---|---|
| 5 | 0 |
| 3x | 1 |
| x² + 2x | 2 |
| 4x³ − x | 3 |
🔹 6. Types of Polynomials (By Number of Terms)
(a) Monomial – One term
এটা পদ থাকিলে → Monomial
Example: 5x²
(b) Binomial – Two terms
দুটা পদ থাকিলে → Binomial
Example: x + 3
(c) Trinomial – Three terms
তিনিটা পদ থাকিলে → Trinomial
Example: x² + 2x + 1
🔹 7. Types of Polynomials (By Degree)
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 7 |
| 1 | Linear | 2x + 1 |
| 2 | Quadratic | x² + 3x + 2 |
| 3 | Cubic | x³ − x |
🔹 8. Value of a Polynomial
To find value, put the given value of variable and calculate.
চলকৰ মান বসাই মান উলিয়াব লাগে।
Example:
Find value of 2x + 3 when x = 2
= 2(2) + 3
= 7
🔹 9. Zero of a Polynomial
If the value of a polynomial becomes zero for some value of x,
that value is called zero of the polynomial.
যি x ৰ মানত বহুপদীৰ মান শূন্য হয় তাক zero কোৱা হয়।
Example:
x − 3 = 0
x = 3 → zero
🔹 10. Graphical Meaning of Zero
The point where the graph cuts the x-axis is the zero of polynomial.
গ্ৰাফে x-axis য’ত কাটে সেই মানেই zero।
🔹 11. Important Points for Exam ⭐
✔ Power of variable must be whole number
✔ Degree = highest power
✔ Linear polynomial → straight line graph
✔ Zero means value becomes 0
🔹 12. Common Mistakes ❌
❌ Writing √x as polynomial
❌ Wrong degree calculation
❌ Forgetting constant has degree 0