The polynomials: Definition, Types, Degree and Properties

If there is one thing that you should pay close attention to during your journey to learn the basics of mathematics, it will undoubtedly have to do with polynomials due to their important role in dealing with different algebraic expressions. Polynomials are included in many mathematical concepts.

In this lesson, we will try to discuss about the definition of the concept of polynomials and its standard form, and we will explain the types of polynomials and the concept of "the degree of polynomial". Finally, we will see some of the properties of the polynomials.

Polynomials and its standard form

Note:

A polynomial is the sum of multiple terms and each term is a mononomial.

Based on this definition, we must first get to know the concept of term in detail because it is the basic building block of which polynomials are composed.

Note:

The term is a product of a number (which is the coefficient) and one or more variables raised to powers. The exponent is a natural integer.

Take, for example:

• The term has coefficient 1.

• The term has coefficient 4.

• The term has coefficient -1.

Like terms in an algebraic expression are terms that consist of the same variable or variables and are raised to the same power.

Take, for example:

• Like terms:

• Unlike terms:

• Unlike terms:

By defining the term, we can now say that a polynomial is the sum of a finite number of terms.

Take, for example:

• is a polynomial in .

• is a polynomial in and .

N.B: The terms of a polynomial cannot have variables in a denominator.

Take, for example:

• is not a polynomial.

Types of polynomials

There are three types to distinguish between types of polynomials.

Monomials: It is an algebraic expression consisting of only one term.

For example,

Binomials: It is an algebraic expression consisting of only two terms.

For example,

Trinomials: It is an algebraic expression consisting of only three terms.

For example,

Binomials and trinomials can also be called polynomials.

Degree of a polynomial

To return to the concept of the term, in the first paragraph above, we said that the term is a number multiplied by a variable or several variables raised to an exponent. So what is the degree of a term?

The degree of the term is the sum of the exponents of the variable(s) that exist in this term.

Take, for example:

• The degree of is .

• The degree of is .

• The degree of is .

• The degree of is .

• More examples

So, what is the degree of a polynomial?

Note:

Degree of a polynomial is the greatest degree of any terms of the polynomial.

Take, for example:

• The degree of is .

• The degree of is .

• More examples

Properties of a polynomials

Now that we know the definition of polynomials and how to determine their degree, we will now move on to some properties about adding, subtracting and multiplying polynomials. In this summary, we will try to take some examples of polynomials with only one unknown in order to simplify the idea of performing the three operations mentioned above.

Note:

The sum or subtraction of two polynomials and is a polynomial .

And write or

The degree of the polynomial is the largest degree of and .

And write

Example1:

Let and are two polynomials in , so that:

Info!

Reminder:

• If the parentheses are preceded by a (+) we remove the parentheses without changing anything.

• If the parentheses are preceded by (- ), we remove the parentheses and then change the signs in those parentheses to their opposite.

• We simplify algebraic expressions using the properties of Combining Like Terms and Distributivity.