One would be hard-pressed to overestimate the importance of polynomials in mathematics -- they literally show up just about everywhere and have an incredible diversity of uses.

Polynomials are built from simpler mathematical objects called monomials -- so let's start there...

A monomial to be the product of a real value called the coefficient and some number of powers of variables whose exponents are non-negative integers. A monomial that has no variable factors is called a constant. The degree of a monomial is the sum of the exponents on its variable powers. Constants -- that have no variable powers present -- have degree zero (since multiplying by a variable to the zero power would not alter the value of that constant).

Consequently, the following are examples of monomials (the first is also a constant)

$$-5,\; x,\; 3p^2,\; m^5,\; \textrm{ and } -\frac{3}{5}ab^3$$

while the expressions below are not...

$$5x+3,\; \frac{2}{p},\; a^3 b^{-2},\;\textrm{ and } \sqrt{x}$$

The table below indicates the coefficients, variables, exponents, and degrees of the monomials given above

$$\begin{array}{|c|c|c|c|c|} \textrm{Monomial} & \textrm{Coefficient} & \textrm{Variables} & \textrm{Exponents} & \textrm{Degree}\\\hline -5 & -5 & \textrm{none} & \textrm{none} & 0\\\hline x & 1 & x & 1 & 1\\\hline 3p^2 & 3 & p & 2 & 2\\\hline m^5 & 1 & m & 5 & 5\\\hline -\frac{3}{5}ab^3 & -\frac{3}{5} & a \textrm{ and } b & 1 \textrm{ and } 3 & 4\\\hline \end{array}$$

Now, with some understanding of monomials at our disposal, we can provide a very simple definition of a "polynomial":

A polynomial is either a monomial or the sum of two or more monomials.

The following are some examples of polynomials:

$$-7, \quad 8x+y, \quad 3x^2 + 2x + 4, \quad \textrm{ and } \quad 5 - 3y + 5xy^2$$

while the below expressions are not

$$x^2 + \frac{3}{x}, \quad x^{-2} + 5x + 3, \quad \textrm{ and } \quad x^2 + 3x^{1/2} + 7$$

Each monomial in a polynomial is called a term of that polynomial.

An individual term may be further described as a "positive term" or a "negative term", in accordance with the sign of the coefficient for that term. For brevity, one should generally write a difference of terms instead of writing a sum of a term and some negative term. For example, instead of writing $x^3 + (-7x^2) + x + (-4)$, we write $x^3 - 7x^2 + x - 4$.

The degree of a polynomial is the greatest degree seen in its terms. Polynomials are classified according to their degree in accordance with the following table:

$$\begin{array}{|c|c|} \textrm{Degree} & \textrm{Referred to as a } \underline{\hspace{0.3in}} \textrm{ polynomial}\\\hline 1 & \textrm{linear}\\\hline 2 & \textrm{quadratic}\\\hline 3 & \textrm{cubic}\\\hline 4 & \textrm{quartic}\\\hline 5 & \textrm{quintic}\\\hline \end{array}$$

We can also classify polynomials by the number of terms they contain. As we have already seen, a polynomial of one term is a monomial. Polynomials of two terms are called binomials, and polynomials of three terms are called trinomials.

Finally, we can classify polynomials by the number of different variables they involve. A polynomial of one variable is a polynomial where all variable powers present in the terms of that polynomial involve the same variable. For example $3x^3 + 5x^2 + 8$ is a polynomial of one variable. In a similar manner, something like $2x^2y - 7y + x - 4$ would be called a polynomial of two variables.

Simplifying Polynomial Expressions

First, unless there is a compelling reason to do otherwise, one should adopt as a standard practice writing polynomials as a sum of terms in descending order. That is to say, the terms of the polynomial are ordered so that the degrees of these terms decrease from left to right. As an example, $x^5 + 7x^2 - 2x + 8$ is written in descending order, while $5x - 2x^4 + 2$ is not.

Second, any "like terms should be collected". Any two monomials which are identical with the possible exception of their coefficients are called like terms. As an example, $2x^2y$ and $3x^2y$ are like terms, while $2x^2y$ and $2x^2y^3$ are not like terms.

We may combine a sum of like terms into a single monomial term by appealing to the distributive property shown below: $$ab + ac = a(b + c) \; \textrm{ and } \; ba + ca = (b + c)a, \; \textrm{ for all real values of } a, b, \textrm{ and } c$$

To see this, consider the sum of like terms $2x^2y$ and $3x^2y$:

$$2x^2y + 3x^2y = (2 + 3)x^2y = 5x^2y$$

This process is called collecting like terms. Note how what we leave in the parentheses only the coefficients of the two terms being added together. These coefficients are by definition real values, so they can be added together to produce a third real value which then plays the role of the coefficient on the resulting monomial term.

Had the terms being summed not been like terms, what would have remained in the parentheses could not have been simplified in the same way, as exampled below

$$2x^2y + 2x^2y^3 = 2x^2y(1+y^2)$$