Zeros of Polynomial
2026-02-28 13:00 Diff
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Last updated on October 23, 2025

The zeros of polynomials are the values of the variables that make the polynomial zero. The values of the variables that make the polynomial 0 are known as zeros of the polynomial. It is also known as the roots of the polynomial, and it is denoted by symbols such as 𝛼, 𝛽, 𝛾, etc. In this article, we will learn about the zeros of a polynomial.

Zeros of Polynomials

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The zeros of polynomials are the values of the variables that make the polynomial zero. 

The values of the variables that make the polynomial 0 are known as zeros of the polynomial. It is also known as the roots of the polynomial, and it is denoted by symbols such as 𝛼, 𝛽, 𝛾, etc.

In this article, we will learn about the zeros of a polynomial.

What are Zeros of Polynomials?

The zeros of a polynomial are the value of x that makes the equation equal to zero. For a function f(x), the zeros of the polynomial are the values of x so that f(x) = 0. If we say that x = a is a zero of the polynomial, it means that if we put x = a in the expression, the answer comes out to be zero. Let’s understand this with the following example,

Let f(x) = x2 - 9

If x = 3

f(3) = 32 - 9 
= 9 - 9 = 0

Therefore, x = 3 is a zero of f(x) = x2 - 9.

Relation between Zeros and Coefficients

In algebra, if we know the zeros of a polynomial, we can find its coefficients, and vice versa.

This relationship is especially useful in quadratic and cubic equations. 

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Relation between Zeros and Coefficient of a Quadratic Equation

If the quadratic equation is in the form of ax2 + bx + c = 0, the two zeros of a quadratic equation are 𝛼 and 𝛽, then 

Sum of roots = 𝛼 + 𝛽 = -b/a

Product of roots = 𝛼 × 𝛽 = c/a

Relation between Zeros and Coefficient of a Cubic Equation

Polynomials are algebraic expressions that include variables, coefficients, powers, and are combined by addition and subtraction. There are different types of polynomials, but two of the most common are:

  • Linear Polynomials
     
  • Quadratic Polynomials

Linear Polynomials: A linear polynomial is in the form of ax + b. The highest degree of the linear polynomial is 1. The zero of a linear formula is found by using a formula: 
x = -ba.

Quadratic Polynomials: The polynomial with the highest degree of 2 is known as a quadratic polynomial. The quadratic polynomial is in the form of ax2 + bx + c. To find the zeros of the quadratic polynomial, we use the formula:
x = -b  D2a

Here, a, b, and c are the numbers from the polynomial.
D is called the discriminant, and it can be found by using the formula,
D = b2 ± 4ac.

How to Find Zero of a Polynomial?

Finding the zeros of the polynomial means solving the equation. There are many ways to find the zero of a polynomial; some of the methods are discussed below:

  • Solving for Linear Polynomial
     
  • Solving for Quadratic Polynomials
     
  • Solving for a Cubic Polynomial
     
  • Solving for Higher Degree Polynomials
     

Solving for Linear Polynomial

Finding zero for a linear polynomial is the easiest way, as it has only one zero. A simple rearrangement of a polynomial can calculate it by setting the polynomial to zero. A linear equation is in the form of y = ax + b, and after simplifying, we will get x = -ba.

Example: Find the zero for the linear polynomial f(x) = 5x - 7.
To find the zero, we have to set f(x) to 0.
5x - 7 = 0
5x = 7
x = 75

Solving for a Quadratic Polynomial

A quadratic equation is in the form of x2 + (a + b)x + ab = 0. We can find the zeros of a quadratic equation in two ways: either by using the factorization or by using the formula.

In factorization, the quadratic equation can be factorized into (x + a)(x + b) = 0, and we have x = -a and x = -b as the zeros of the polynomial. We can also find the zeros using the formula:\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Solving for Cubic Polynomial

The cubic polynomial is in the form of y = ax3 + bx2 + cx + d. By using the following steps, we can find the zero of a cubic polynomial.

Step 1: Try simple values like x = 1, 2, -1, -2 in the equation.

Step 2: If we get y = 0, then that number is the root or zero of the polynomial.

Step 3: Divide the polynomial by the root to get a quadratic polynomial.

Step 4: Solve for the quadratic polynomial by using the formula.

Solving for Higher Degree Polynomials

To find the zeros of higher-degree polynomials, we can follow these steps:

Step 1: Try for small values like 1, 2, 3, …  in the equation using the remainder theorem to check if they are roots

Step 2: If the value makes the polynomial equal to zero, then that is the root.

Step 3: Divide the polynomial by that root to reduce it.

Step 4: Contin+ue the process till the polynomial becomes a quadratic polynomial.

Step 5: Solve the quadratic polynomial.

Sum and Product of Zeros of Polynomial

When we solve a polynomial, we find its roots or zeros. We don’t need to solve the polynomial to find the sum and product of the polynomial.

In this section, we will learn how to find the sum and product of the zeros of a polynomial. 
 

Sum and Product of Zeros of a Polynomial for Quadratic Equation

The sum and product of a quadratic polynomial can be calculated from the variables of the quadratic equation without finding the zeros of the polynomial. 𝛼 and 𝛽 are used to represent the zeros of the quadratic polynomial.

The sum and product of zeros of the polynomial are as follows:

Sum of Zeros of Polynomial = 𝛼 + 𝛽 = -b/a

Product of Zeros of Polynomial = 𝛼𝛽 = c/a

Sum and Product of Zeros of Polynomial for Cubic Equation

The general form of a cubic polynomial is ax3 + bx2 + cx + d = 0. Here 𝛼, 𝛽, and 𝛾 are used to represent the roots of a cubic polynomial. 

𝛼 + 𝛽 + 𝛾 = -b/a

𝛼 × 𝛽 × 𝛾 = d/a

𝛼𝛽 + 𝛼𝛾 + 𝛽𝛾 = c/a

Zeros in Graph of Polynomials

The zeros of a polynomial are the x-values at which the graph intersects the x-axis. The x-coordinates of those points are the values that make the polynomial equal to zero, i.e., f(x) = 0.

A polynomial expression can be linear, quadratic, or cubic based on the degree of the polynomial. The graph of the zeros of polynomials is given below:

Common Mistakes and How to Avoid Them in Zeros of Polynomial

When learning about the zeros of polynomials, it is easy to get confused, especially with signs, formulas, or graphs etc. These small mistakes can lead to wrong answers. In this section, we will learn some common mistakes and the ways to avoid them to master the zeros of polynomials. 

Real Life Applications of Zeros of Polynomial

Zeros of polynomials play a major role in real-life applications such as engineering, physics, economics, etc. Understanding and finding the zeros of polynomials helps us model and solve real-life problems more effectively. 

  • Physics: In physics zeros of a polynomial are used to find when a moving object reaches a certain point. For example, to predict when a ball thrown upwards reaches the ground. 
     
  • Engineering: In engineering, we use the zeros of a polynomial to model vibration, structural loads, and stress analysis. 
     
  • Computer graphics and animation: Polynomial curves are used in computer graphics. Zeros of polynomials help in collision detection, to find where curves intersect or objects meet. 
     
  • Astronomy: In astronomy, polynomial equations are used to model planetary orbits and satellite paths. The zeros help to find when two paths intersect or when an object comes to rest. 

Problem 1

Find the zero of the linear polynomial: f(x) = 2x - 6

Okay, lets begin

x = 3

Explanation

Set the polynomial to 0.

2x - 6 = 0

Simplify the equation.

2x = 6

x = 3
 

Well explained 👍

Problem 2

Find the zeros of a quadratic polynomial: f(x) = x2 - 5x + 6

Okay, lets begin

 x = 2 and x = 3

Explanation

Factor the quadratic expression:

x2 - 5x + 6 = (x - 2)(x - 3)

Now, set each factor to 0.

x - 2 = 0, x - 3 = 0

x = 2, x = 3

Well explained 👍

Problem 3

Find the zeros of the quadratic polynomial: f(x) = x2 + 4x + 5.

Okay, lets begin

No real zeros

Explanation

Use the formula,

 \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

 \(x = {-4 \pm \sqrt{16-20} \over 2}\)


\(x = {-4 -42}\)

x = -4  2i2

x = -2 ± i

Since the square root of a negative number is imaginary, this polynomial has no real zeros. 
 

Well explained 👍

Problem 4

Find the zero of a cubic polynomial: f(x) = x3 - 4x2 + 5x - 2.

Okay, lets begin

x = 1, 2

Explanation

Try x = 1


f(1) = 1 - 4 + 5 - 2 = 0


So, x = 1 is one of the zeros of the given polynomial.


Now divide f(x) by (x - 1):(x3 - 4x2 - 3x + 2) ÷ (x - 1)
= x2 -3x +2


Factorize x2 - 3x + 2:


x2 - 3x + 2 = (x - 1)2(x - 1)


So the zeros are, x = 1, 2

Well explained 👍

Problem 5

Find the zeros of f(x) = 3x2 - 12

Okay, lets begin

 x = 2, x = -2

Explanation

Set the polynomial to 0

f(x) = 3x2 - 12 = 0

3x2 = 12

x2 = 4

x = 2

Well explained 👍

FAQs on Zeros of Polynomials

1.What are the zeros of polynomials?

The values of the variable for which the polynomial becomes zero are known as the zeros of the polynomial.

2.Are the zeros and roots of a polynomial the same?

Yes, the terms zeros, roots, and solutions of a polynomial refer to the same values that make the polynomial equal to zero. 

3. How many zeros can a polynomial have?

The number of zeros of a polynomial is based on its degree; that is, if the degree is n, then the number of zeros is n. These may be real or complex, and some may be repeated.

4.Can a polynomial have repeated zeros?

Yes, the zeros of a polynomial can repeat.

5. What does it mean graphically when a polynomial has a zero?

The zero of the polynomial is where the graph crosses or touches the x-axis. 

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.