Monomials | Binomials | Trinomials | Polynomials

The Monomials, Binomials and Trinomials are the types of Polynomials differentiated based on the number of terms present in the Polynomial.

Monomials

Definition of Monomials

The word “Mono” means one. Therefore, the Monomials can be defined as Algebraic Expressions that have only one term. The Monomials may contain a single constant or variables or a combination of both constants and variables.

For the Monomials having variables, the degree of the Monomial should be a positive Integer. The algebraic expressions that have only one term but the degree of the term is negative are not Monomials.

Examples of Monomials

The examples of different types of Monomials are as follows:

Monomials

Monomials having only a Single Constant

The examples of Monomials having only one constant are:

1\,, 2\,, 3\,, 4\,, 12\,, 45\,, -1\,, -4\,, -5\,, \frac{3}{2}\,, \frac{5}{9}\,, \frac{-1}{6}\,,\frac{-9}{14}\,, etc.

Monomials having a Single Variable

In an Algebraic Expression, the variables are denoted by letters. Therefore, examples of some Monomials that has only one variable are:

x\,, y\,, z\,, p\,, q\,, r\,, a\,, b\,, c\,, etc.

Monomials having two or more Variables

Some examples of Monomials having two or more variables are:

xy\,, yz\,, xz\,, -xyz\,, ab\,, ca\,, abc\,, pq\,, qr\,, etc.

Monomials having Combination of Constants and Variables

As mentioned above, Monomials may be a combination of both constants and variables. Some examples of such Monomials are:

4x\,, 5y\,, 2z\,, 9xy\,,14xz\,,-3xy\,,-5yz\,, etc.

In the above examples, all the algebraic expressions have only one term and the degree of each term is positive. Therefore, all of the above terms are Monomials.

Degree of a Monomial

The Degree of a Monomial is the highest power of any variable present in the Monomial. As discussed above, the degree of a Monomial must be a Positive Integer. If the Degree of a term is a Negative Number or a Fraction, then the term is not a Monomial.

To understand the Degree of a Monomial, let us consider the following examples:

  • 4x^3

In the above example of Monomial, the variable is x and the power of x is 3. Therefore, the Degree of the above Monomial is 3.

  • 6xy^4

In the above example, there are two variables i.e., x and y. The power of x is 1 and the power of y is 4. The degree of the term is 1+4=5, which is a positive integer. Therefore, the degree of the above Monomial is 5.

  • xy^{-3}

Similarly, in the above example, there are two variables x and y. The power of x is 1 and y is -3. Hence, the degree of the term is 1+(-3)=-2. As the degree of the term is negative, therefore, the term is not a Monomial.

Properties of Monomials

Some properties of Monomials are as follows:

  • The multiplication of a Monomial with another Monomial gives a Monomial.
  • The division of a Monomial with another Monomial may be a Monomial or may not be a Monomial.
  • A Monomial Multiplied by a Monomial result a Monomial.
  • A Monomial Multiplied by a Constant also results in a Monomial.
  • The addition of two Monomials gives a Binomial.
  • Subtraction of two Monomials also gives a Binomial.

Binomials

Definition of Binomials

The meaning of the word “Bi” is two. Therefore, the Binomials may be defined as the Algebraic Expressions that have precisely two terms. The two terms may be separated by addition or subtraction.

For an Algebraic Expression to be a Binomial, first the Algebraic Expression must have two terms and second, the Exponent or Power of variables in both the terms must be a positive integer. If the Exponent or Power of any variable in any term is negative or a fraction, then the Algebraic Expression is not a Binomial even if it has two terms.

Binomial

Examples of Binomials

Some examples of Binomials are described below:

  • 3x + 4y

The above Algebraic Expression has two terms 3x and 4y and the Power of the variable in the first term is 1 and in the second term is also 1. Since, both the terms have positive powers, therefore, the above Algebraic Expression is a Binomial.

  • x – y^2

Similarly, in the above Algebraic Expression, there are two terms and the Power of the variable in the first term is 1 and in the second term is 2 which are positive integers. Therefore, the above Algebraic Expression is a Binomial.

  • x^3+z^{-2}

For the above Algebraic Expression, the Power of the variable z in the second term is -2, which is a Negative Integer. Therefore, the above Algebraic Expression is not a Binomial.

Trinomials

Definition of Trinomials

The word “Tri” means three. Therefore, the Trinomials may be defined as the Algebraic Expressions that have exactly three terms. The three terms may be separated by addition or subtraction.

It is to be noted that for an Algebraic Expression to be a Trinomial, first, there must be three terms present in the Algebraic Expression and second, the Exponent or Power of Variables in each term must be a Positive Integer. If the power of any variable in any term is negative or fraction, then the Algebraic Expression is not considered as a Trinomial.

Trinomial

Examples of Trinomials

Some examples of Trinomials are discussed below:

  • x+y^2+z^3

For the above example, the Algebraic Expression has three terms and the Power of Variables in all the three terms is a Positive Integer. Therefore, the above Algebraic Expression is a Trinomial.

  • x^2+3xy-y^{-2}

Similarly, in the above example, the Algebraic Expression also has three terms. However, the Power the variable y in the third term is -2 which is a Negative Integer. Therefore, the above Algebraic Expression is not a Trinomial.

Polynomials

Definition of Polynomials

The meaning of the word “Poly” is many. therefore, the Algebraic Expressions that have one or more than one term are called Polynomials. Thus, a Monomial, Binomial and Trinomial are all Polynomials. It is also to be noted that for an Algebraic Expression to be a Polynomial, the Power of the variables present in any term of the Algebraic Expression must be a Positive Integer. Otherwise, the Algebraic Expression can not be considered as a Polynomial.

Examples of Polynomials

Some examples of Polynomials are as follows:

  1. \displaystyle {{x}^{4}}+4{{x}^{3}}+3{{x}^{2}}+2x+1
  2. \displaystyle {{x}^{3}}+5{{x}^{2}}+3x+2
  3. \displaystyle -{{x}^{2}}+4x+2

Frequently Asked Questions (FAQ)

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The word "Mono" means one. Therefore, the Monomials are the Algebraic Expressions that have only one term.

The word "Mono" means one. Therefore, the Monomials can be defined as Algebraic Expressions that have only one term. The Monomials may contain a single constant or variables or a combination of both constants and variables.
The example of some Monomials are: 1, 2, 3, a, b, c, x, y, z, x², y³ etc.

Some properties of Monomials are as follows:
1. The multiplication of a Monomial with another Monomial gives a Monomial.
2. The division of a Monomial with another Monomial may be a Monomial or may not be a Monomial.
3. A Monomial Multiplied by a Monomial result a Monomial.
4. A Monomial Multiplied by a Constant also results in a Monomial.
5. The addition of two Monomials gives a Binomial.
6. Subtraction of two Monomials also gives a Binomial.

The meaning of the word "Bi" is two. Therefore, the Binomials may be defined as the Algebraic Expressions that have precisely two terms. The two terms may be separated by addition or subtraction.

A Binomial is called so because it has precisely two terms. The meaning of the word 'Bi' is two.

The word "Tri" means three. Therefore, the Trinomials may be defined as the Algebraic Expressions that have exactly three terms. The three terms may be separated by addition or subtraction.

The Degree of a Monomial is the highest power of any variable present in the Monomial. As discussed above, the degree of a Monomial must be a Positive Integer. If the Degree of a term is a Negative Number or a Fraction, then the term is not a Monomial.
To understand the Degree of a Monomial, let us consider the following example,
4x³
In the above example of Monomial, the variable is x and the power of x is 3. Therefore, the Degree of the above Monomial is 3.

The Monomials can be defined as Algebraic Expressions that have only one term. The Monomials may contain a single constant or variables or a combination of both constants and variables.

The Binomials may be defined as the Algebraic Expressions that have precisely two terms. The two terms may be separated by addition or subtraction.

If an Algebraic Expression has only one term then it is called a monomial. A monomial will not have any addition on subtraction operator.

Yes, 5x² is a monomial because it has only one term.

If an Algebraic Expression has precisely two terms then it is a Binomial. The two terms of a Binomial will always be separated by addition or subtraction operator.

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