Not every algebraic expression is a polynomial! For an expression to be a true polynomial, the variable exponents must be whole numbers only ($0, 1, 2, 3\dots$). Negative powers (like $x^{-1} = 1/x$) or fractional roots (like $x^{1/2} = \sqrt{x}$) disqualify an expression from being a polynomial.
Standard form in one variable $x$: $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ where $a_n \ne 0$. The highest exponent $n$ is the Degree.
| Degree | Classification | Standard Form | Example | Number of Zeroes |
|---|---|---|---|---|
| $0$ | Constant Polynomial | $p(x) = c$ ($c \ne 0$) | $5, -12$ | Zero zeroes |
| $1$ | Linear Polynomial | $ax + b$ ($a \ne 0$) | $3x - 7$ | At most 1 zero |
| $2$ | Quadratic Polynomial | $ax^2 + bx + c$ | $2x^2 + 5x - 3$ | At most 2 zeroes |
| $3$ | Cubic Polynomial | $ax^3 + bx^2 + cx + d$ | $x^3 - 2x^2 + 1$ | At most 3 zeroes |
Note on Zero Polynomial: The polynomial $p(x) = 0$ has not defined degree!
Problem: Check whether $p(x) = x + \frac{2}{x}$ is a polynomial.
Analysis: Rewrite with exponents: $p(x) = x^1 + 2x^{-1}$.
Conclusion: The exponent of the second term is $-1$, which is not a whole number. Hence, $p(x)$ is NOT a polynomial.
In $\sqrt{2}x^2 + 3x + 5$, students often claim it is not a polynomial because of the square root!
Fact: The radical is on the constant coefficient $\sqrt{2}$, NOT on the variable $x$. The variable powers are $2$ and $1$ (whole numbers). Hence, it is a perfectly valid polynomial.
Game physics engines compute projectile bullet drop and vehicle skid trajectories using quadratic polynomials ($s = ut - \frac{1}{2}gt^2$) updated sixty times every second.