Zeros of a Function

The zeros of a function f(x) are the values of x that make f(x) = 0. The values of x are known as the roots of a function. Graphically, the zeros are the points where the function crosses or touches the x-axis, also called x-intercepts.

There are numerous methods to find the zeros of a function. These different methods include:

• Graphical Method

• Factorization Method

• Quadratic Formula Method

• Newton-Raphson Method

• Bisection Method

• Graphical Method: The zeros of a function are the value of x that makes f(x) = 0. The graphical method can be used to find the zeros of the function. The zeros are points where the graph of the function intersects the x-axis. 

• Factorization Method: In the factorization method, to find the zeros, we convert the function into simple factors. So we first factor the function and set each factor equal to zero, and solve them.  

           For example: f(x) = x² + 7x + 10
           It can be factorized into: f(x) = (x + 2)(x + 5)
           Setting each factor equal to zero: x + 2 = 0 and x + 5 = 0
           Finding the value of x:
           x + 2 = 0
           x = -2

           x + 5 = 0
           x = -5
          Therefore, the zeros of the function f(x) = x² + 7x + 10 are: 
           x = -2 and x = -5. 

• Quadratic Formula Method: The quadratic formula is used to find the root of a quadratic equation. For any equation in the form: f(x) = ax2 + bx + c, where a ≠ 0. The quadratic formula is:

       x = [-b ± √(b² - 4ac)] / 2a

• Newton-Raphson Method: To find the roots of a real-valued function, we use the Newton-Raphson method. After the famous scientists, Sir Isaac Newton and Joseph Raphson, this method is named. This method starts with an initial guess x0 and gradually improves it through successive iterations to approach the actual root of the function. 

           Assume x1 = x0 + h, where x0 is the approximate root of the equation
           f(x1) = 0, so f(x0 + h) = 0 

           Using Taylor’s theorem, expanding f(x0 + h):
           f(x0) + hf′(x0) + …. = 0
           Then h = -fx0f'x0
            So, x₁ = x₀ - f(x₀)/f'(x₀)
            xn + 1 = xn - fxnf'xn 

• Bisection Method: The bisection method is used to find the roots of a polynomial or continuous function within a specific interval. It is used when the function changes sign over the intervals, that is, f(a)∙f(b) < 0, stating that the root lies between a and b. Let’s now learn how the bisection method works step by step:

           Assume the points a and b such that a < b and f(a)∙f(b) < 0. 
           Next, calculate the midpoint of a and b, so m = (a + b)/2
           The next interval is selected based on the sign of f(m), that is:

           If f(m) = 0, then m is the root

           If f(m) < 0, choose the interval from m to b

           If f(m) > 0, choose the interval from a to m. 
 

The zeros of a function f(x) represent the solutions to the equation: f(x) = 0. In other words, we are finding the value of x, which makes f(x) = 0. For the value of x, different methods like grouping, algebraic identities, splitting the middle term, etc., are used.

How to Find Real Zeros of a Function?

The real zeros of a function f(x) are real numbers r that make f(x) = 0. In other words, it is a value of x for which the function equals zero. 
For example, f(x) = 3x3 - 6x2 - 9x
= 3x(x2 - 2x - 3)
Factoring x2 - 2x - 3 as (x - 3)(x + 1)
So, f(x) = 3x(x - 3)(x + 1)
3x(x - 3)(x + 1) = 0
x - 3 = 0 → x = 3
x + 1 = 0 → x = -1
x = 0
So, the real zeros of the function f(x) = 3x3 - 6x2 - 9x are x = 0, x = 3, and x = -1.

How to Represent Zeros of a Function on a Graph

We find the zeros of a function using a graph; that is, the point where the graph intersects the x-axis is the root of the function. Here, we will learn how to find the value of the root of f(x2 - 4) using the graph. 

The graph is of the function: f(x) = x2 - 4. Here, the graph intersects the x-axis at two points (2 and -2), so x = 2 and x = -2. Therefore, 2 and -2 are the roots (zeros) of the function that make f(x) = 0. 
 

The zeros of a function are used in real life in the fields of physics, engineering, computer security, mathematics, etc. In this section, we will learn the applications of the zeros of a function. 

• The zero of a function is used to show how the function behaves over a certain range; that is, by knowing where the function is zero, we can determine intervals where the function is positive or negative. 

• For solving problems related to equilibrium and physical systems, zeros in the fields of engineering and physics are used to determine equilibrium points. 

• In mathematics, we use zeros to solve differential equations, polynomial equations, etc.