Polynomials

A polynomial is the algebraic sum of monomials. The monomials that make up a polynomial are called terms of the polynomial.

The polynomials are among the most important objects in mathematics and form the basis of mathematical analysis, algebra and analytical geometry.

This lesson is a starting point on polynomials and serves to introduce the basic concepts that will be useful in future lessons.

In particular, in this lesson we will see what a polynomial is, how to reduce a polynomial to normal form and how to find the degree of a polynomial.

We will also give the definitions of binomial, trinomial and quadrinomial and see when two polynomials are equal.

Polynomials

In the lesson on operations with monomials we saw that the sum of two monomials does not always result in another monomial. Let's consider the following sum, for example:

In such case, in the sum, all monomials are similar. So we can add all monomials together getting a single monomial as a result:

Conversely, let us try to consider the following expression:

In this case the monomials are not similar. Therefore the result will not be a monomial:

This is referred to as Polynomials:

Starting with the definition, let's see whether the following expressions are polynomials or not.

For example, the following expressions are polynomials:

Conversely, the following expressions are not polynomials:

The two expressions above are not polynomials because there are operations between the monomials other than algebraic addition.

From the definition, we can also infer that every monomial is also a polynomial. This is because we can rewrite a monomial as the sum of itself and the null monomial. For example:

Furthermore, every number is a polynomial, since every number is also a monomial. The zero, , in particular is called the null polynomial as well as null monomial.

Normal form of a polynomial

A polynomial can be made up of similar terms, i.e. similar monomials. For example, the following polynomial contains monomials that are similar to each other:

In this case, similar monomials can be added together:

What we obtain is called a polynomial in normal form.

Let's look at another example. The following polynomial is not reduced in normal form:

To reduce it to normal form, we first identify all similar monomials:

Once the similar monomials have been identified, we sum them together:

Binomials, Trinomials and Quadrinomials

For certain polynomials reduced to normal form, a special name is given:

Here are a few examples:

• Binomials:

• Trinomials:

• Quadrinomials:

Equality between polynomials

From the normal form of a polynomial, we can also define when two polynomials are equal:

For example, the following two polynomials are equal to each other:

This is because the terms of which they are composed are the same even if they appear in a different order.

Degree of a polynomial

Starting from the normal form of a polynomial, it is possible to define the degree of a polynomial:

It follows from the definition that in order to find the degree of a polynomial, we must first find the degrees of its terms, i.e. the degrees of its monomials.

Let's clarify the concept with an example, looking at the following polynomial reduced to normal form:

The polynomial consists of 5 terms:

As we can see, the third term, , has the highest degree, , so the polynomial is of fourth degree or of degree .

As with monomials, it is also possible to define the degree with respect to a letter for polynomials:

Returning to the example above, we can derive the degrees with respect to the letters of the individual terms:

In the example, the term of maximum degree with respect to is the third one: , degree 3 with respect to . On the other hand, the term of maximum degree with respect to is the fourth: , degree 3 with respect to .

This is why the polynomial is of degree 3 with respect to , and is also of degree 3 with respect to .

Known term of a Polynomial

Some polynomials reduced to normal form may contain a monomial of degree 0, i.e. a number. This term is called the known term:

For example, in the following example, is the known term:

Homogeneous Polynomial

From the degree of a polynomial, we can define homogeneous polynomials:

For example, let us look at the following polynomial:

It is a homogeneous polynomial as all the monomials in it are of degree 2.

Ordered Polynomial

For example, the following polynomial is ordered with respect to in ascending order:

Conversely, the following polynomial is ordered with respect to in descending order:

Complete Polynomial

For example, the following polynomial is complete with respect to :

This polynomial is of degree 3 with respect to x and has, moreover, all powers of , starting with and ending with . In fact, the known term, 7, can be rewritten as: .

In brief

This lesson provides the basis for the study of polynomials.

Here, we have introduced the concept of a polynomial, which represents the algebraic sum of monomials.

Next, we saw what normal form is and how to reduce a polynomial to normal form. Starting with the normal form, we defined binomials, trinomials and quadrinomials.

Normal form is necessary to define the equality between two polynomials and the degree of a polynomial.

All these concepts will be fundamental in the next lessons where we will study the operations that can be carried out on polynomials.