Types of Polynomials (Based on Terms and Degrees)

Types of Polynomials: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. There are mainly four types of polynomials based on degree-constant polynomial (zero degree), linear polynomial ( 1st degree), quadratic polynomial (2nd degree), and cubic polynomial (3rd degree).

There are 3 types of polynomials based on the number of terms in the polynomial - monomial, binomial, and trinomial, and for more than that we use the general term polynomial. This article is about the types of polynomials - Monomials, Binomials, and Polynomials in detail.

Types of Polynomial

Polynomials can be categorized based on the number of terms they contain. Each category has unique characteristics and applications in mathematics.

• Monomial
• Binomial
• Trinomial

Monomial

An algebraic expression that contains only one non-zero term is known as a monomial. A monomial is a type of polynomial, like, a binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero. It consists of only a single term which makes it easy to do the operation of addition, subtraction, and multiplication.

Examples:

• g is a monomial in one variable.
• 9cb² is a monomial in two variables c and b.
• 3a²b is monomial in two variables a and b.
• 4ab/5m is monomial in three variables a, b, m.
• -2m is a monomial in one variable m.

The different parts of a monomial expression are:

• Variable: The letters present in the monomial expression are variables.
• Coefficient: The number before a variable or the number multiplied by a variable in the expression.
• Literal Part: The alphabets which are present along with the exponent values are the literal part.

Monomial Examples - 6xy²

• Coefficient is 6
• Variables are x and y
• Degree of monomial expression = 1 + 2 = 3
• The literal part is xy²

Monomial Degree

The sum of exponent values of variables in the expression is called the degree of monomial or monomial Degree. If variables don't have any exponent values its implicit value is 1.

Example:

4xy³, In this exponent value of x is 1. 

degree of expression is 3 + 1 = 4.

Monomial Operations 

The arithmetic operations which are performed on the monomial expression are addition, subtraction, multiplication, and division.

Addition of Two monomials:

When we add two monomials with the same literal part, it will result in a monomial expression. 

Example:

Addition of 2xy + 4xy = 6xy

Subtraction of Two monomials:

When we subtract two monomials with the same literal part, it will result in a monomial expression.

Subtraction of 6ab - 4ab = 2ab.

Multiplication of Two monomials:

When we multiply two monomials with the same literal part, it will result in a monomial expression.

Product of 2a²b * 6x = 12a²bx

Division of Two monomials:

When we divide two monomials with the same literal part, it will result in a monomial expression.

Division of x⁶ by x² = x⁴

Binomial

An algebraic expression that contains two non-zero terms is known as a binomial. It is expressed in the form ax^m + bxⁿ where a and b number, x is variable, m and n are nonnegative distinct integers.

Examples:

• g + 3m is a binomial in two variables g and m.
• 3a⁴ - 5b²  is a binomial in two variables a and b.
• -4x² - 9y is a binomial in two variables x and y.
• a²/4 + b/2 is a binomial in two variables a and b.

Binomial Equation

Any equation that contains one or more binomials is known as a binomial equation.

Example:

v = u + 1/2 at²

Operations on Binomials

A few basic operations on binomials are 

• Factorization
• Addition
• Subtraction
• Multiplication
• Raising to the nth power
• Converting to lower-order binomials

Factorization:

A binomial can be expressed as the product of the other two.

Example: 

a² - b² can be expressed as (a + b) (a - b).

Addition:

Two binomials can be added if both contain the same variable and the same exponent.

Example:

(2a² + 3b) + (4a² + 5b) = 6a² + 8b

Subtraction: 

It is similar to addition, two binomials should contain the same variable and exponent.

Example:

(6a² + 3b) - (2a² + 5b) = 4a² - 2b

Multiplication:

When we multiply two binomials distributive property is used and it ends up with four terms. In this method, multiplication is carried by multiplying each term of the first factor to the second factor.

Example:

(ax + b) (mx + n) can be expressed as amx² + (an + mb) x + bn

Raising to nth Power: 

A binomial can be raised to the nth power and expressed in the form of  (x + y)ⁿ

Converting to Lower order binomials:

Higher-order binomials can be factored to lower-order binomials such as cubes can be factored to products of squares and another monomial.

Example:

a³ + b³ can be expressed as (a + b) (a² - ab + b²).

Polynomial

An algebraic expression that contains one, two, or more terms is known as a polynomial.

Examples: 

• 3a + 4b is a polynomial of two terms a and b.
• 2a³ + 3b² + 4m - 5x + 6k  is a polynomial of five terms in five variables .
• a + 2a² + 3a³ + 4a⁴ + 5a⁵ + 6a⁶  is a polynomial of six terms in one variable.

Types of Polynomials

• Monomial: An algebraic expression that contains only one non-zero term is known as a monomial. A monomial is a type of polynomial, like, a binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero.
• Binomial: An algebraic expression that contains two non-zero terms is known as a binomial. It is expressed in the form ax^m + bxⁿ where a and b number, x is variable, and m and n are nonnegative distinct integers.
• Trinomial: An algebraic expression that contains three non-zero terms is known as the Trinomial. For example, a + b + c is a trinomial in three variables a, b, and c.

Degree of a Polynomial

In the polynomial equation, the variable having the highest exponent is called the degree of the polynomial. 

Example:

3a⁵ + 4a³ - 2a + 6

The degree of above polynomial is 5.

Polynomial Equations

The standard form of representing a polynomial equation is to put the highest degree first and the constant term at last.

Example: 

x⁴ + 2x³ + 3x² + x + 5

Solving Polynomials

We can easily solve polynomials using basic algebra and factorization concepts, generally, while solving polynomials' the first step is to set the right-hand side to 0.

Solving Linear Polynomial:

1. The first step is to isolate variable term
2. Next, make the equation equal to 0
3. Solve it using basic algebra operations.

Example: Solve 4a - 8?

Solution:

4a - 2 = 0

=> 4a = 8

=> a = 8 / 4

=> a = 2

Solving Quadratic Polynomial:

1. The first step is to rewrite the expression in descending order of degree.
2. Next, equate it to 0
3. Perform polynomial factorization.

Example: Solve 4a² - 4a + a³ - 16?

Solution:

Rearranging, a³ + 4a² - 4a - 16

=> a³ + 4a² - 4a - 16 = 0

=> a² (a + 4) - 4 (a + 4) = 0

=> (a + 4) (a² - 4) = 0

Solution is a = -4 and a² = 4

People Also Read:

• Polynomials Meaning, Identities, Degree, Types, Examples
• Polynomial Formula
• Polynomial Functions
• Polynomial Identities: Definition, Types, Proof, and Examples
• Degree of a Polynomial

Operations on Types of Polynomials

Multiplication of Monomials

Example: Multiply 4a and 3ba³?

Solution: 

First we need to group Coefficients and Variables

(4 * 3) (a * a³) (b)

Apply exponential law,

12a¹⁺³b

12a⁴b

Multiplication of three or more monomials

Example: Multiply a², 2ab³, 4ab?

Solution: 

(4 * 2) (a² * a * a) (b³ * b)

8a⁴b⁴

Multiplication of monomial by binomial

Example: Multiply 2a by a + 4?

Solution: 

(2a * a) + (2a * 4)

2a² + 8a

Multiplication of monomial by trinomial 

Example: Multiply 3a by 2a² + 3ab + 4?

Solution: 

(3a * 2a²) + (3a * 3ab) + (3a * 4)

6a³ + 9a²b + 12a

Multiplication of Binomial by a Binomial

Example: Multiply 4a + 3 and 2a +1?

Solution: 

4a (2a + 1) + 3 (2a + 1)  [ now its like multiplication of monomial and binomial ]

8a² + 4a + 6a + 3

8a² + 10a + 3

Multiplication of Binomial and Trinomial 

Example: Multiply 4a + 1 and a² + 2a + 1?

Solution: 

4a (a² + 2a + 1) + 1 (a² + 2a + 1)

4a² + 8a + 4a + a² + 2a + 1

5a² + 14a + 1

Multiplication of Polynomial and Monomial 

Example: Multiply a³ + a² + a + b + 3 and 4a?

Solution: 

(4a * a³) + (a² * 4a) + (a * 4a) + (b * 4a) + (3 * 4a)

4a⁴ + 4a³ + 4a² + 4ab + 12a

Multiplication of Polynomial and Polynomial

Example: Multiply 2x⁴ + 3x⁵ + 4 and 2x + 1?

Solution: 

(2x⁴ * 2x) + (3x⁵ * 2x) + (4 * 2x) + (2x⁴ * 1) + (3x⁵ * 1) + (4 * 1)

2x⁵ + 6x⁶ + 8x + 2x⁴ + 3x⁵ + 4

6x⁶ + 5x⁵ + 2x⁴ + 8x + 4

Practice Problems on Types of Polynomials

1. Determine the type of the polynomial P(x) = 4x³ - 2x² + 7x - 5.

2. Classify the polynomial Q(x) = x⁴ - 3x³ + 2x² by its degree and by the number of terms.

3. What is the degree of the polynomial R(x) = 7x⁵ - 2x³ + x² + 4.

4. Add the following polynomials and identify the type of the resulting polynomial

V(x) = 2x² + 3x + 1

W(x) = -x² + 4x -2