Breaking Down the Definition
The word itself hints at the meaning: "poly" is Greek for "many," and "nomial" traces back to the Latin word for "term," so a polynomial is, quite literally, a sum of many terms. Written out for a variable x, a general polynomial takes this shape:
Each a is a fixed real number, called a coefficient, and n must be a non-negative whole number - 0, 1, 2, 3, and so on. That restriction is exactly what separates a polynomial from other kinds of expressions: no fractional exponents, no negative exponents, and no variable sitting under a radical or in a denominator.
Naming Polynomials by Degree
The degree of a polynomial is just the highest exponent attached to its variable, and it's usually the first thing you're asked to identify. Here's how the common degrees are named:
| Polynomial | Degree | Name |
|---|---|---|
| 7 | 0 | Constant |
| 3x + 1 | 1 | Linear |
| x² - 4x + 4 | 2 | Quadratic |
| 2x³ + x | 3 | Cubic |
| x⁴ - 1 | 4 | Quartic |
Writing a Polynomial in Standard Form
Standard form means the terms are arranged from the highest degree down to the lowest. It's not required mathematically, but it makes a polynomial far easier to read, compare, and divide later on:
x - 2x² + 4x³ - 8 ✗ (not standard)
Adding and Subtracting Polynomials
Adding or subtracting polynomials just means combining terms that share the same degree - everything else is left alone:
= (3+1)x² + (2-5)x + (-1+4)
= 4x² - 3x + 3
Multiplying Polynomials
To multiply two polynomials, pair every term in the first one with every term in the second, add up all the products, and then simplify by combining any matching terms:
= x³ - 3x² + x + 2x² - 6x + 2
= x³ - x² - 5x + 2
What Are the Roots (or Zeros) of a Polynomial?
A root, sometimes called a zero, is any x-value that makes the entire polynomial equal zero. A polynomial of degree n can never have more than n real roots. Factoring the expression is the usual way to find them, and for a quadratic specifically, the quadratic formula gets you there directly.