Concept Notes: Polynomial Definition, Coefficients, Degree & Standard Form
1. Rigorous Definition of a Polynomial in One Variable
An algebraic expression of the form:
$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$
is a polynomial in $x$, provided:
- The coefficients $a_n, a_{n-1}, \dots, a_0$ are real numbers ($\mathbb{R}$).
- The leading coefficient $a_n \neq 0$.
- Every exponent of $x$ is a non-negative integer (whole number: $0, 1, 2, 3, \dots$).
Figure 2.1: Polynomial Classification — Identifying Non-Negative Integer Exponents, Coefficients and Degree Hierarchies
2. Identifying Valid vs Invalid Polynomials
| Expression | Is it a Polynomial? | Reason |
|---|---|---|
| $3x^2 - 5x + 7$ | YES | Exponents are 2, 1, 0 (all whole numbers). |
| $4x + \frac{2}{x}$ | NO | $\frac{2}{x} = 2x^{-1}$; exponent $-1$ is a negative integer! |
| $\sqrt{5}x + 8$ | YES | Coefficient $\sqrt{5}$ is real; exponent of $x$ is $1$. |
| $3\sqrt{x} + 4$ | NO | $\sqrt{x} = x^{1/2}$; exponent $1/2$ is not an integer! |
| $y^3 - \sqrt{2}y + 11$ | YES | Exponents are 3, 1, 0 (all whole numbers). |
3. Classification by Degree and Number of Terms
- By Degree (Highest Power of Variable):
- Constant Polynomial (Degree 0): $P(x) = c = cx^0$ (e.g., $P(x) = 9$).
- Zero Polynomial: $P(x) = 0$. Degree is undefined.
- Linear Polynomial (Degree 1): $\mathbf{P(x) = ax + b \quad (a \neq 0)}$.
- Quadratic Polynomial (Degree 2): $P(x) = ax^2 + bx + c$ ($a \neq 0$).
- Cubic Polynomial (Degree 3): $P(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$).
- By Number of Non-Zero Terms:
- Monomial: Exactly 1 term (e.g., $5x, -7$).
- Binomial: Exactly 2 terms (e.g., $2x + 3, x^2 - 4$).
- Trinomial: Exactly 3 terms (e.g., $x^2 + 5x + 6$).