Each power of \(x\), e.g., \(x^{1}, x^{2}, x^{n}\), etc. is called **monomial** in \(x\).

A monomial times a constant coefficient, e.g., \(3x^{2}\) is called a **term** of the polynomial.

****WARNING****

This terminology is not entirely uniform; in particular some books define a ``monomial'' to be what we have defined to be a ``term,'' and vice-versa, and some make no distinction.

**Definition:**

The **degree**, \(N,\) of a polynomial \(P(x)\), is the degree of the highest order term such that \(c_N x^{N} \neq 0\).

**Example:**

A polynomial of degree zero has the equation

\[P(x) = c_0. \]

I.e. it is a constant function.

**Example:**

A polynomial of degree one has the equation

\[ P(x) = c_0 + c_1x.\]

It is the equation of a straight line with:

- \(y\)-intercept \(c_0\),
- slope \(c_1\),
- and \(x\)-intercept \(= \alpha = - \frac{c_0}{c_1}.\)

**Example Problem:**

Are there any polynomials in \(x\) that are also linear functions of \(x\)? **Solution: **

The only polynomial in \(x\) that is also a linear function of \(x\) is the monomial \(P(x) = c_1x\).