What is a Polynomial?
A polynomial is an algebraic expression that contains one or more terms, where each term consists of a variable raised to a non-negative integer exponent, multiplied by a coefficient (a real number).
The word "polynomial" comes from "poly" (many) and "nomial" (terms).
Key Components of a Polynomial:
| Component | Definition | Example in 3x² + 2x - 5 |
|---|---|---|
| **Coefficient** | The numerical factor of a term | 3, 2, -5 |
| **Variable** | The letter representing an unknown quantity | x |
| **Exponent** | The power to which the variable is raised | 2, 1, 0 |
| **Degree** | The highest exponent in the polynomial | 2 |
Polynomials in One Variable:
A polynomial in one variable (say x) has the general form:
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₀ are constants (coefficients).
Types of Polynomials Based on Degree:
| Type | Degree | Standard Form | Example |
|---|---|---|---|
| **Linear** | 1 | ax + b | 3x + 4, -2x + 1 |
| **Quadratic** | 2 | ax² + bx + c | x² - 5x + 6, 2x² + 3x - 1 |
| **Cubic** | 3 | ax³ + bx² + cx + d | x³ - 8, 2x³ + x² - x + 4 |
Types of Polynomials Based on Number of Terms:
| Type | Number of Terms | Example |
|---|---|---|
| **Binomial** | 2 | x + 3, 2x² - 5, 4x³ + 1 |
| **Trinomial** | 3 | x² + 5x + 6, 3x³ - 2x² + x |