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
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.
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?
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.
Adding and subtracting of a two polynomials
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.
Let's Adding P and Q:
Let's Subtracting P and Q:
So as you can see in the two operations, is also a polynomial and its degree is 3 ( has degree 3 and has degree 2, so, has degree 3).
Let's Multiplying P and Q:
So as you can see, is also a polynomial and its degree is 5 ( has degree 3 and has degree 2, so, has degree 3+2=5).
Info!
For the division, we will try to explain it in another lesson because it is not quite as simple as the above-mentioned operations.