Polynomial
AlgebraA polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents.
Formula
a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0
Definition
A polynomial is a math expression made by adding or subtracting terms that have variables with whole-number exponents. Each piece is called a term.
Example
$3x^2 + 5x - 2$ is a polynomial with three terms. $x^4 - 7$ is also a polynomial. The exponents must be whole numbers ($0, 1, 2, 3, \ldots$).
Key Insight
Polynomials are the building blocks of algebra. Most equations you study in algebra involve polynomials.
Definition
A polynomial in one variable $x$ is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where the $a_i$ are real coefficients and the exponents are non-negative integers. The degree is the highest exponent. Polynomials are closed under addition, subtraction, and multiplication.
Example
$4x^3 - 2x^2 + x - 9$ is a degree-$3$ polynomial. Adding $(2x^2 + 3)$ and $(4x^2 - x)$ gives $6x^2 - x + 3$, still a polynomial.
Key Insight
Polynomials cannot have negative exponents, fractional exponents, or variables in the denominator. These restrictions define what makes an expression a polynomial versus a rational or radical expression.
Definition
The set of polynomials over a field $F$ in variable $x$, denoted $F[x]$, forms a commutative ring (actually a principal ideal domain). Every polynomial of degree $n$ over the complex numbers has exactly $n$ roots (counted with multiplicity) by the Fundamental Theorem of Algebra. Polynomial rings are the algebraic setting for Galois theory, which classifies which polynomial equations can be solved by radicals.
Example
$\mathbb{Z}[x]$ (polynomials with integer coefficients) is a UFD but not a field. Gauss's Lemma states that a primitive polynomial in $\mathbb{Z}[x]$ is irreducible in $\mathbb{Z}[x]$ iff it is irreducible in $\mathbb{Q}[x]$.
Key Insight
Polynomials generalize to multivariate polynomial rings $F[x_1,\ldots,x_n]$, the central objects of algebraic geometry. Varieties are zero sets of polynomial ideals, connecting algebra to geometry.