Zeros of Polynomial

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.

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.

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. 

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

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.

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
   

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

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

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.

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.

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. 
 

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

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

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:

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.