Polynomial Basics

Degree, leading coefficient, constant term, roots — the vocabulary of polynomials.

The idea

A polynomial in $x$ is a sum of multiples of whole-number powers of $x$:

$p(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0.$

The numbers $a_n, \ldots, a_0$ are its coefficients. The powers must be whole numbers, which rules out $x^{-1}$, $\sqrt{x}$ (which is $x^{1/2}$), and $2^{x}$ — none of these is a polynomial in $x$.

The degree is the highest power of $x$ whose coefficient is not zero. The leading coefficient is the coefficient of that highest power. The constant term is $a_0$, the term with no $x$, and it equals $p(0)$. A polynomial is monic when its leading coefficient is $1$. A root (also called a zero) of $p$ is a number $x_0$ with $p(x_0) = 0$.

The small degrees have names of their own: degree $0$ is constant, degree $1$ is linear, degree $2$ is quadratic, and degree $3$ is cubic.

Ways to work on it

• Walkthrough. Definition of polynomial, degree, leading coefficient, monic, roots.
• Practice. Read off degree / leading coefficient / constant term from a random polynomial.
• Hardest. Test the definition, find the roots of a difference of squares, and name polynomials by degree.

Not sure where to start? Take the ten-question placement test.