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

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.

For example, consider the function \(f(x) = x^{2} - 4\)

To find its zeros, we set \(f(x) = 0:\)
\(x^2 - 4 = 0\\ x^2 = 4\\ x = \pm 2\)

​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: \(f(x) = x^2 + 7x + 10\)
It can be factorized into: \(f(x) = (x + 2)(x + 5)\)
Setting each factor equal to zero: \(x + 2 = 0 {\text { and }} x + 5 = 0 \)
Finding the value of x: 

\(x + 2 = 0\\ x = -2\\ \\ \ \\ x+ 5 = 0\\ x = -5\\ \)
        
Therefore, the zeros of the function \(f(x) = x^2 + 7x + 10\) are \(x = -2 {\text { and }} x = -5\)

​

The quadratic formula is used to find the root of a quadratic equation. For any equation in the form: \(f(x) = ax^2 + bx + c\), where\( a ≠ 0\). The quadratic formula is:
       \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

For example, solving \(2x^2 + 3x - 2 = 0\)

Here, \(a = 2, b = 3, c = -2\)

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

Substituting the value: \(x = {-3 \pm \sqrt{3^2-4(2)(-2)} \over 2(2)}\)

Simplifying: \(x = {-3 \pm \sqrt{9+16} \over 4}\\ x = {-3 \pm \sqrt{25} \over 4}\\ x = {-3 \pm 5 \over 4} \\\)
Solving both cases: 

\(x = {{{-3 + 5} \over 2}} = {{2\over 4}} = 0.5 \\ x = {{{-3 - 5} \over 2}} = {{-8\over 4}} = -2 \\ \)

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 \(x_0\) and gradually improves it through successive iterations to approach the actual root of the function. 
 

Assume \(x_1 = x_0 + h\), where \(x_0\) is the approximate root of the equation
\(f(x_1) = 0, {\text { so }} f(x_0 + h) = 0 \)

Using Taylor’s theorem, expanding \(f(x_0 + h)\):
\(f(x_0) + hf′(x_0) + …. = 0 \)
Then \(h = {{-f(x_0) \over f'(x_0)}}\)
 So, \(x_1 ={{ x_0 - {f(x_0)\over f'(x_0)}}}\)
\(x_{n + 1} ={{ x_n - {f(x_n)\over f'(x_n)}}}\)

For example, find the \(\sqrt{2}\) using Newton-Raphson Method. 

\({x^2} = 2 \implies {f(x) = x^2 - 2} = 0\)

So, \(f(x) = x^2 - 2, {\text { }} f′(x)=2x\)

Applying the formula: \(x_{n + 1} ={{ x_n - {f(x_n)\over f'(x_n)}}}\)

 Let \(x_0 = 1.5\)
\(x_{1} ={{ 1.5 - {(1.5^2 - 2)\over 2(1.5)}}}\\ \\ \ \\ = 1.5 - {{0.25 \over 3}}\\ \\ \ \\ = 1.4167\)
 

Let \(x_1 = 1.4167\)
\(x_{2} ={{ 1.4167 - {(1.4167^{2} - 2)\over 2(1.4167)}}}\\ \\ \ \\ = 1.4167 - {{0.0069 \over 2.8334}}\\ \\ \ \\ = 1.4142\)

Let \(x_2 = 1.4142\)
\(x_{3} ={{ 1.4142 - {(1.41427^{2} - 2)\over 2(1.4142)}}}\\ \\ \ \\ = 1.4142\)

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, \(f(a)\cdot f(b) < 0\), 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 \(f(a)\cdot f(b) < 0\). 
Next, calculate the midpoint of a and b, so \(m = {{(a + b)\over2}}\)
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. 

For example, find the root of the function: \(f(x) = x^2 -4\)
As \({{f(x) = 0 }} \) when \(x^2 = 4\), so the root should be x = 2 or x = -2 

Choose an interval [a, b]:

let a = 1 and b = 3

\(f(1) = 1^2 - 4 = -3\) (negative)

\(f(3) = 3^2 - 4 = 5\) (positive)

Find the midpoint: \(m = {{a + b} \over 2}\)
\(= {{1 + 3} \over 2}\\ = 2 \\ \ \\ f(2) = 2^2 - 4 = 0\)

Since f(2) = 0

so, x = 2

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.

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, \(f(x) = 3x^3 - 6x^2 - 9x \)
\(= 3x(x^2 - 2x - 3) \)
Factoring \(x^2 - 2x - 3 {\text { as }} (x - 3)(x + 1)\)

\({\text {So, }}f(x) = 3x(x - 3)(x + 1)\\ \\ \ \\ 3x(x - 3)(x + 1) = 0\\ \\ \ \\ x - 3 = 0 → x = 3\\ \\ \ \\ x + 1 = 0 → x = -1\\ \\ \ \\ x = 0 \)
 

So, the real zeros of the function \(f(x) = 3x^3 - 6x^2 - 9x {\text { are }} x = 0, x = 3, {\text { and }} x = -1\).

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 \(f(x^2 - 4)\) using the graph. 

The graph is of the function: \(f(x) = x^2 - 4\). 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 \(f(x) = 0\). 
 

Understanding the zeros of a function is fundamental concept to analyze the behavior and solving equations. In this section, we will learn a few tips and tricks to master it.

 

• Remember that zeros are the x-values that make the function equal to zero. So always start to setting the equation equal to 0 and solving for x.
  
   

• Factorize the function if possible; is the function is a polynomial, factor it completely. For example, \({{f(x) = x^2 - 9}} = {{(x - 3) (x + 3)}}\). So, the zeros are \({x = 3 {\text { and }} x = - 3}\).
  
   
• Use a graph to visualize the function; the points where the graph crosses or touches the x-axis represent the zeros. 
  
   
• After finding zeros, don't forget to substitute it back into the original function to confirm that \({f(x) = 0}\).
  
   
• Memorize the quadratic formula, that is, \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\). Use it when the polynomial is non-factorable.  

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 determine 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 represent equilibrium points, where forces balance out, such as in mechanical systems, circuits, or motion analysis.

• In mathematics, Zeros are used to find solutions of polynomial and differential equations, which form the basis of many mathematical models.

• In coding theory and data encryption, zeros of polynomial functions are used to design error-detecting and error-correcting codes that ensure secure data transmission.

• In business, zeros are used to identify break-even points, where profit equals cost showing when a company neither makes a loss nor a gain.