Types of Polynomials

Types of Polynomials
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We have been studying polynomials from a very young age.

When we were first told about variables and expressions, we were simply dealing with polynomials.

For example, if Mathew has \(x\) pencils and if John has 2 more pencils than Mathew, then John must have \(x + 2\) pencils.

\(x + 2\) is a polynomial.

Two children holding pencils

But do you know that polynomials can be classified into different types?

For example, Mathew has \(x\) pencils, which is a monomial and John has \(x+2\) pencils, which is a binomial.

In this mini-lesson, we will explore the world of polynomials by learning about different types of polynomials through interesting simulations, some solved examples, and a few interactive questions for you to test your understanding.

Let's begin!

Lesson Plan

What Are Polynomials?

Polynomials are expressions with one or more terms having a non-zero coefficient.

These terms comprise variables, exponents, and constants.

A standard polynomial is one where the highest degree is the first term, and the subsequent terms are arranged in ascending order based on the powers or the exponents.

Polynomials are classified based on the number of terms present in the expression as a monomial, binomial, and trinomial, which we will see in detail under types of polynomials.

Other terms related to polynomials are constant, coefficient, and degree of the polynomial. 

The first term of the polynomial is called the leading term.

The number without any variable is called a constant.

The number preceding a variable is called a coefficient.

The power of the leading term or the highest power of the variable is called the degree of the polynomial.

Terms related to a polynomial


What Are the Types of Polynomials Based on Degrees?

Based on the degree of the polynomial, polynomials can be classified into 4 major types:

  • Zero or Constant polynomial
  • Linear polynomial
  • Quadratic polynomial
  • Cubic polynomial

Look at the table given below to understand the meaning of the types of polynomials with examples.

Type of polynomial Meaning Examples
Zero or constant polynomial Polynomials with 0 degrees are called zero polynomials.

\(3\ or\ 3x^0\)

\(56\ or\ 56y^0\)

\(8\ or\ 8z^0\)

Linear polynomial Polynomials with 1 as the degree of the polynomial are called linear polynomials. In linear polynomials, the highest exponent of the variable(s) is 1

\(x+y-4\)

\(5m+7n\)

\(2p\)

Quadratic polynomial Polynomials with 2 as the degree of the polynomial are called quadratic polynomials.

\(8x^2+7y-9\)

\(m^2+mn-6\)

\(2p^2-7\)

Cubic polynomial Polynomials with 3 as the degree of the polynomial are called cubic polynomials.

\(3x^3\)

\(6m^3-mn+n^2-4\)

\(p^3+pqr-7\)

 
tips and tricks
Tips and Tricks
  1. Before learning about the types of polynomials, it is important to learn what do we consider as polynomials.
  • Polynomials should have a whole number as the degree. Expressions with negative exponents are not polynomials. For example, \(x^{-2}\) is not a polynomial.
  • Polynomials do not have variables in their denominator. For example, \(\dfrac{2}{x+2}\) is not a polynomial.

What Are the Types of Polynomials Based on Terms?

There are different types of polynomials. Based on the number of terms, polynomials are classified as:

  • Monomials
  • Binomials
  • Trinomials

A monomial is a polynomial expression that contains only one term.

For example, \(2x + 5x + 10x\) is a monomial because when we add the like terms it results in \(17x\).

Furthermore, \(4t\), \(21x\), \(2y\), \(9pq\) are also monomials.

A binomial is a polynomial with two, unlike terms.

For example, \(3x + 4x^2\) is a binomial as it contains two unlike terms, that is, \(3x\) and \(4x^2\). 

Likewise, \(10pq + 13p^2\) is also a binomial.

A trinomial is a polynomial with three, unlike terms.

For example, \(3x + 5x^2 – 6x^3\) is a trinomial.

Likewise, \(12pq + 4x^2 – 10\) is also a trinomial.

We can be classifying polynomials based on the number of terms into 3 types.

But, we can also have more than 3 terms in the polynomials.

Polynomials that have 4 unlike terms are called four-term polynomials.

Similarly, polynomials with 5 terms are called five-term polynomials, and so on.

 
important notes to remember
Important Notes
  1. In a polynomial, two terms are separated by addition or subtraction sign. Multiplication and division operators are not used to create more terms in a polynomial. For example, we consider \(3\times x\times y\) as one term and \(3x+y\) as two terms in a polynomial.
  2. Based on the terms in a polynomial, it can be classified into the following 3 types:

    Monomial

    Binomial

    Trinomial

  3. Based on the degree of a polynomial, it can be classified into 4 types:

    Zero polynomial
    Linear polynomial
    Quadratic polynomial
    Cubic polynomial

Solved Examples 

Example 1

 

 

Find the volume of the given cube if the length of each side is \(x\) units. 

Also, state the type of polynomial obtained.

cube of side x units

Solution

We know that volume of a cube = \((side)^3\)

Here, the side-length of the square is given as \(x\) units.

Hence, the volume of the given cube = \(x^3\)

\(x^3\) has only one term with degree 3. 

\(\therefore\) It is a monomial cubic polynomial.

\(\therefore\) Volume of the given cube = \(x^3\). This is a monomial cubic polynomial.
Example 2

 

 

Classify the given polynomials.

classify the given polynomials - polynomials are shown on a green board

Solution

There are 5 polynomials given on the board.

Let's classify each one of them.

\(n^3+6\) is a binomial cubic polynomial as the highest exponent (degree of polynomial) with the variable is 3 and there are 2 terms in the polynomial.

\(5x^2-2xy+1\) is a trinomial quadratic polynomial as the degree is 2 and there are 3 terms in the polynomial.

\(p-82\) is a binomial linear polynomial as the degree of the polynomial is 1 and there are 2 terms in the polynomial.

\(2p^2+q-11\) is a trinomial quadratic polynomial. The degree of the polynomial is 2 and there are 3 terms present in it.

\(34\) is a monomial zero polynomial as the degree of the polynomial is 0 and there is a single term in the polynomial.


Interactive Questions on Types of Polynomials

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of types of polynomials. The math journey around types of polynomials starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Types of Polynomials

1. What are the 3 types of polynomials?

Based on the number of terms in a polynomial, there are 3 types:

  • Monomial
  • Binomial
  • Trinomial

2. How do you identify a polynomial?

A polynomial is an expression in which the terms containing coefficients and variables are connected through four basic operators, addition, subtraction, multiplication, and division.

In a polynomial, there should not be any negative number as the exponent of the variable. The variable should not be in the denominator of the polynomial that is in fractional form.

3. How do you classify polynomials?

Polynomials can be classified on the basis of the number of terms and on the basis of the degree of the polynomial. 

Based on the number of terms in a polynomial, it can be classified into 3 types:

  • Monomial
  • Binomial
  • Trinomial

Based on the degree of a polynomial, it can be classified into 4 types:

  • Zero polynomial
  • Linear polynomial
  • Quadratic polynomial
  • Cubic polynomial

4. What is a polynomial with 5 terms called?

Polynomials with 5 terms are known as five-term polynomials. For example, \(3x^3-xy+y^2+6x^2-8\).

5. What is a polynomial with 4 terms called?

Polynomials with 4 terms are known as quadrinomials. For example, \(x^3+3x-2y+5\).

6. What is a polynomial with a degree of 2 called?

Polynomials with degree 2 are called quadratic polynomials.

7. What is a degree 5 polynomial?

A polynomial with degree 5 is called a quintic polynomial. For example, \(2x^5+x^4y-3y^2+7y-4\).

8. What is a degree 6 polynomial?

Polynomial with degree 6 is known as hexic or sextic polynomial. One example of a hexic polynomial is \(p^6-3pq\).
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