Polynomials are widely used algebraical objects. They have the form of a sum of scaled powers of a variable. 

Monomials over $\mathbb{R}$

Let $\mathbb{R}$ be the set of real numbers. A monomial over $\mathbb{R}$ in a single variable $x$ consists of a non-negative power of $x$, multiplied with a nonzero constant $c \in \mathbb{R}.$ So a polynomial looks like

$cx^n$,

where $n \geq 0$ is an integer and $c \not = 0$ is a real number. If we want to give the polynomial a name, say $M$, we denote that its variable is $x$ by writing $x$ between brackets:

$M(x)=cx^n$.

The exponent $n$ is called the degree of $M(x).$ The constant $c$ is the coefficient.

Examples

$\sqrt{2}x^7$is a monomial of degree 7 and coefficient $\sqrt{2}$. 

$7x^{\sqrt{2}}$, $\sqrt{2}x^{-7}$ and $2x^7 - 7x^2$are not monomials. The first and the second do not have a non-negative integer exponent and the third is a sum of two monomials.

Polynomials over $\mathbb{R}$

A polynomial over $\mathbb{R}$ is a finite sum of monomials over $\mathbb{R}$. For example 

$P(x)= 4x^{13} +3x^2-\pi x + 1$

is the finite sum of the $4$ monomials: $4x^{13}, 3x^2, -\pi x$ and $1 = 1x^0.$ 

It is also the sum of the 6 monomials: $1/3 x^{100}, -1/3 x^{100}, 4x^{13}, 3x^2, -\pi x$ and $1$, as will be explained in the discussion about addition and subtraction of polynomials. However, we can only write down $P(x)$ as the sum of monomials of distinct degree in exactly one way, namely the first we mentioned. These monomials are called the terms of $P(x).$The coefficients of $P(x)$ are the coefficients corresponding to its terms.

Every monomial is also a polynomial, as it can be written as a sum with one term, itself. 

A special example of a polynomial is the zero polynomial

$Z(x) = 0,$

which is a sum of $0$ monomials. 

The degree of a polynomial $Q(x)$ is the highest degree of one of its terms. For example, the degree of $P(x)$ is $13$.

The degree of the zero polynomial is defined to be $-\infty$.

Extra: Polynomials Over General Rings

This part is for the interested reader only. Most students can skip this part, or just remember that polynomials over $\mathbb{C}$ are the same as polynomials over $\mathbb{R}$, but with complex coefficients and that the degree of a monomial in more variables equals the sum of the exponents.

We have discussed polynomials over $\mathbb{R}$. We shall later see that we can add, subtract and multiply these polynomials. In general, our coefficients $c$ do not need to belong to $\mathbb{R}$, but they can belong to any set of "numbers" in which we can add, subtract and multiply. These sets are called rings. Examples of rings are the real numbers $\mathbb{R}$, the integers $\mathbb{Z}$ and the complex numbers $\mathbb{C}$. In this case, we talk about complex polynomials, or polynomials over $\mathbb{C}$. The degree of a polynomial is defined in the same way as in the real case.

In particular, the polynomials over $\mathbb{R}$ form a ring, which we denote by $\mathbb{R}[x]$. The polynomials over this ring will be polynomials in two variables $x$ and $y$ over $\mathbb{R}$.

Here the degree in $x$ of $x^3y^5$is $3$, the degree in $y$ of $x^3y^5$is $5$ and its joint degree or degree is $8$.