Polynomials in One Variable
2026-02-28 08:55 Diff
https://brightchamps.com/en-us/math/algebra/polynomials-in-one-variable

201 Learners

Last updated on December 10, 2025

Algebraic expressions are polynomials made up of variables and coefficients, and combined using arithmetic operations. Polynomials in one variable have the form axn, where a is a real number and n is a non-negative integer.

What are Polynomials in One Variable?

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

Polynomials in one variable are algebraic expressions consisting of terms with single variable and coefficients, combined using arithmetic operations. For example:

x2 + 5x + 8

2y3 + 5y2 + 9y + 3  

The above examples have only one variable (‘x’ in the first example and ‘y’ in the second).

Classification of Polynomials in One Variable

Polynomials in one variable can be grouped into different types, depending on their degree. The degree of a polynomial is simply the highest exponent of the variable. Based on this, polynomials are usually divided into four main types: 

  • Zero or Constant Polynomial
  • Linear Polynomial
  • Quadratic Polynomial
  • Cubic Polynomial 

Zero or Constant Polynomial: A zero or constant polynomial’s variable has no exponent, meaning its degree is 0. As a result, these polynomials only include constant terms. For example, 2x0 = 21 = 2.

Linear Polynomial: The polynomials with 1 as the highest degree of the variable are the linear polynomials. For example, x + 2, 6x + 5. 

Quadratic Polynomial: Quadratic polynomials are the polynomials with the highest degree of 2. For example, 5x2 + 2x + 6, 6y2 + 5y + 1

Cubic Polynomial: A cubic polynomial is one in which the highest degree of the polynomial is 3. For example, 5x3 + 2x2 + 8x + 7
 

How to Solve Polynomials in One Variable?

Solving a polynomial is the process of finding the values of the variable that make the whole expression equal to zero. The solution of the polynomial is also known as roots or zeros of the polynomial. A polynomial of degree n can have up to n roots. So, linear polynomials have one root, quadratic polynomials have two roots and cubic polynomials have three roots. 

Solving Linear Polynomial in One Variable
 

The general form of a linear polynomial is ax + b = 0. Now let’s learn how to solve a linear polynomial, with an example:

3x - 6 = 0

Step 1: To solve, isolate the term with the variable

3x = 6

Step 2: Isolate the variable

x = 6/3

x = 2

Explore Our Programs

Solving Quadratic Polynomial in One Variable

Factoring: In the factoring method, the quadratic polynomial is expressed as a product of two binomial expressions. The general form of a quadratic equation is ax2 + bx + c = 0. To factor, we find two numbers whose product is equal to ac and sum is equal to b. 

For example, finding the root of x2 + 5x + 6 = 0

Here, a = 1, b = 5, and c = 6

Here the factors are 2 and 3, whose sum is 5 and product is 6

So, x2 + 5x + 6 = (x + 2)(x + 3) = 0

x + 2 = 0 and x + 3 = 0

x = -2 and x = -3

Using Quadratic Formula: The general form of a quadratic polynomial is ax2 + bx + c = 0, where a ≠ 0. The formula to find the value of x is

For example, 2x2 - 4x - 6 = 0

Here, a = 2, b = -4 and c = -6

x = 3 and x = -1

Solving Cubic Polynomial in One Variable

The general form of a cubic equation is ax3 + bx2 + cx + d = 0, where a ≠ 0. To solve a cubic equation, follow these steps: 

For example, finding the value of x in

Here, the equation is in standard form:

If

So, x = 1 is a root

Factoring:

Solving the quadratic equation:

So, x = 1, 2, 3

Tips and Tricks of Polynomials in One Variable

Polynomials may look tricky, but they’re actually patterns made of numbers and variables. A polynomial in one variable means it has only one type of variable, like x or y. These handy tips and tricks will help you easily identify, simplify, and understand polynomials in one variable.
 

  • A polynomial in one variable means it has only one variable (like x, y, or z).
  • The highest power of the variable tells you the degree of the polynomial.
  • Arrange terms from the highest power to lowest for clarity.
  • Combine only terms with the same variable and power.
  • The number before the variable is called the coefficient.

Common Mistakes and How to Avoid Them in Polynomials in One Variable

When learning polynomials in one variable, students often get confused with the concept and make errors. Here are a few common mistakes that we can avoid in the future. 

Real-World Applications of Polynomials in One Variable

Polynomials in one variable are useful in various fields, such as business, science, and engineering, for modeling situations, making predictions, and analyzing patterns. In this section, we will learn some real-world applications of polynomials. 

  • In physics, polynomials in one variable are used in modeling projectile motion. It helps to calculate the trajectory of a ball, a rocket, or a jet of water from a fountain. 
  •  In civil engineering, a polynomial in one variable is used to design roller coasters to ensure smooth and continuous curves to ensure a safe and exciting ride for riders. 
  • In finance, to calculate the compound interest over several periods, financial planning, retirement saving projections, and loan calculations we use the polynomials in one variable. 
     

Problem 1

Find the degree of polynomial 8x2 + 6x + 5x3 - 6

Okay, lets begin

The degree of a polynomial is 3
 

Explanation

 The degree of a polynomial is based on the largest power of the variable. In this case, the highest power is 3, so the polynomial is of degree 3.
 

Well explained 👍

Problem 2

Identify the type of polynomial based on their degree: 5x^2 + 6x, x - 5

Okay, lets begin

 Here, is a quadratic polynomial and x - 5 is a linear polynomial
 

Explanation

 Based on the degree, the polynomials are classified into linear, quadratic, and zero degree polynomials. 

In , the highest degree of polynomial is 2, so it is a quadratic polynomial.

In x - 5, the highest degree of polynomial is 1, so it is a linear polynomial.

Well explained 👍

Problem 3

Find the root of the polynomial: 2x - 6

Okay, lets begin

 x = 3
 

Explanation

 Arranging the equation in general form ax + b = 0

Well explained 👍

Problem 4

Which of the following are polynomials in one variable? 2x2 + 5x + 5xy 3y3 + 5y - 6y2 x + 2

Okay, lets begin

Here, and are polynomials in one variable. 
 

Explanation

 The first expression, has two variables (x and y). Hence, it is not a polynomial in one variable. 
 

Well explained 👍

FAQs on Polynomials in One Variable

1.What is a polynomial in one variable?

An algebraic expression that contains only one variable is called a polynomial in one variable.
 

2.List a few examples of polynomials in one variable?

Some examples of polynomials in one variable are: 3x + 7, 5x2  + 6x - 7, 6x2 + 9x + 3x3. 
 

3.What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of its variable. For example, in 5x3 + 8x2 + 6x, the degree of the polynomial is 3 as 3 is the highest value of the exponent. 
 

4.What are the types of polynomials based on degree?

Zero or constant, linear, and quadratic polynomials are different types of polynomials based on their degree. 
 

5.Is 5 a constant polynomial?

Yes, 5 is a constant polynomial, because it can be written in the form 5x0. 

6.What should parents know about polynomials in one variable to help their child?

Parents should know that a polynomial in one variable is an expression with only one variable (like x), raised to different powers.

7.How can parents make learning polynomials interesting for their child?

Parents can use real-life examples for instance, comparing each term in a polynomial to items in a store. Only similar items (like terms) can be added together.

8.How can parents explain the “degree” of a polynomial to their child?

Parents can tell their child that the degree is simply the highest power of the variable.
For example, in , the degree is 3 because x3 has the highest exponent.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.