How to find the zeros of a function?
There are different ways to find the zeros of a function. In this discussion, we will learn the best 3 methods of them.
But first, we have to know what are zeros of a function (i.e., roots of a function).
Table of Contents - What you will learn
What are zeros of a function
The zeros of a function f(x) are the values of x for which the value the function f(x) becomes zero i.e. f(x)=0.
Consequently, we can say that if x be the zero of the function then f(x)=0.
To understand the definition of the roots of a function let us take the example of the function y=f(x)=x.
Here the value of the function f(x) will be zero only when x=0 i.e. f(0)=0.
How to find the zeros of a function
We will learn about 3 different methods step by step in this discussion.
Finding the zeros of a function by Factor method
In this method, first, we have to find the factors of a function. Then we equate the factors with zero and get the roots of a function.
Example 1: how do you find the zeros of a function x^{2}+x-6.
For zeros, we first need to find the factors of the function x^{2}+x-6.
The factors of x^{2}+x-6 are (x+3) and (x-2).
Now we equate these factors with zero and find x
i.e., x+3=0 and x-2=0
i.e., x=-3 and x=2.
In a simple way,
x^{2}+x-6=0
or, x^{2}+(3-2)x-6=0
or, x^{2}+3x-2x-6=0
or, x(x+3)-2(x+3)=0
or, (x+3)(x-2)=0
Either (x+3)=0 or (x-2)=0
i.e., either x=-3 or x=2.
Therefore the zeros of a function x^{2}+x-6 are -3 and 2.
Example 2: Find the zeros of the function x^{3} - 4x^{2} - 9x + 36.
x^{3} - 4x^{2} - 9x + 36 = 0
or, x^{2}(x - 4) - 9(x - 4) = 0
or, (x - 4) (x^{2} - 9) = 0
or, (x - 4) (x - 3) (x + 3) = 0
Either x - 4 = 0 or x - 3 =0 or x + 3 = 0
Either x = 4 or x = 3 or x = -3
Therefore the zeros of the function x^{3} - 4x^{2} - 9x + 36 are 4, 3 and -3.
You can watch this video (duration: 5 min 47 sec) where Brian McLogan explained the solution to this problem.
Finding the zeros of a function by solving an equation
There are some functions where it is difficult to find the factors directly.
For these cases, we first equate the polynomial function with zero and form an equation. Then we solve the equation. The roots of an equation are the roots of a function.
Suppose the given polynomial is f(x)=2x+1 and we have to find the zero of the polynomial.
Now equating the function with zero we get,
2x+1=0
or, 2x=-1
or, x=- \frac{1}{2}
Therefore the zero of the polynomial 2x+1 is x=- \frac{1}{2}.
Example: Find the root of the function \frac{x}{a}-\frac{x}{b}-a+b.
First, we equate the function with zero and form an equation. Then we solve the equation and find x.
\frac{x}{a}-\frac{x}{b}-a+b=0
or, \frac{x}{a}-\frac{x}{b}=a-b
or, \frac{bx-ax}{ab}=a-b
or, \frac{x(b-a)}{ab}=-\left ( b-a \right )
or, x(b-a)=-ab\left ( b-a \right )
or, x=\frac{-ab(b-a)}{(b-a)}
canceling (b-a) we get,
or, x=-ab
There the zeros or roots of a function is -ab.
Read also: Best 4 methods of finding the Zeros of a Quadratic Function
How to find the zeros of a function on a graph
This method is the easiest way to find the zeros of a function.
In this method, we have to find where the graph of a function cut or touch the x-axis (i.e., the x-intercept).
The points where the graph cut or touch the x-axis are the zeros of a function.
Now look at the examples given below for better understanding
Question: How to find the zeros of a function on a graph y=x.
Here the graph of the function y=x cut the x-axis at x=0.
Therefore the roots of a function f(x)=x is x=0.
Question: How to find the zeros of a function on a graph g(x) = x^{2} + x - 2
The graph of the function g(x) = x^{2} + x - 2 cut the x-axis at x = -2 and x = 1.
Therefore the roots of a function g(x) = x^{2} + x - 2 are x = -2, 1.
The graphing method is very easy to find the real roots of a function.
But some functions do not have real roots and some functions have both real and complex zeros.
One such function is q(x) = x^{2} + 1 which has no real zeros but complex.
Now the question arises how can we understand that a function has no real zeros and how to find the complex zeros of that function.
To understand this concept see the example given below
Question: How to find the zeros of a function on a graph q(x) = x^{2} + 1
The graph of the function q(x) = x^{2} + 1 shows that q(x) = x^{2} + 1 does not cut or touch the x-axis.
So the function q(x) = x^{2} + 1 has no real root on x-axis but has complex roots.
If we solve the equation x^{2} + 1 = 0 we can find the complex roots.
x^{2} + 1 = 0
or, x^{2} = - 1
or, x = \pm \sqrt{-1}
or, x = \pm \: i
or, x = + \: i,\: - \: i
Therefore the roots of a function q(x) = x^{2} + 1 are x = + \: i,\: - \: i .
Sometimes it becomes very difficult to find the roots of a function of higher-order degrees.
In these cases, we can find the roots of a function on a graph which is easier than factoring and solving equations.
Question: How to find the zeros of a function on a graph h(x) = x^{3} – 2x^{2} – x + 2
After plotting the cubic function on the graph we can see that the function h(x) = x^{3} - 2x^{2} - x + 2 cut the x-axis at 3 points and they are x = -1, x = 1, x = 2.
Therefore the roots of a polynomial function h(x) = x^{3} - 2x^{2} - x + 2 are x = -1, 1, 2.
Question: How to find the zeros of a function on a graph p(x) = \log_{10}x
From the graph of the function p(x) = \log_{10}x we can see that the function p(x) = \log_{10}x cut the x-axis at x= 1.
So the roots of a function p(x) = \log_{10}x is x = 1.
Watch this video (duration: 2 minutes) for a better understanding
Frequently Asked Questions on zeros or roots of a function
The roots of the quadratic equation are 5, 2 then the equation is
As the roots of the quadratic function are 5, 2 then the factors of the function are (x-5) and (x-2).
Multiplying these factors and equating with zero we get,
\: \: \: \: \: (x-5)(x-2)=0
or, x(x-2)-5(x-2)=0
or, x^{2}-2x-5x+10=0
or, x^{2}-7x+10=0,
which is the required equation.
Therefore the quadratic equation whose roots are 5, 2 is x^{2}-7x+10=0.The number of the root of the equation is equal to the degree of the given equation – true or false?
It is true that the number of the root of the equation is equal to the degree of the given equation.
It is not that the roots should be always real. Sometimes we can’t find real roots but complex or imaginary roots.
For example this equation x^{2}=4\left ( y-2 \right ) has no real roots which we learn earlier.What is the number of polynomial whose zeros are 1 and 4?
Zero of a polynomial are 1 and 4.
So the factors of the polynomial are (x-1) and (x-4).
Multiplying these factors we get,
\: \: \: \: \: (x-1)(x-4)
= x(x-4) -1(x-4)
= x^{2}-4x-x+4
= x^{2}-5x+4,
which is the required polynomial.
Therefore the number of polynomials whose zeros are 1 and 4 is 1.
We hope you understand how to find the zeros of a function.
We have discussed three different ways. If you have any doubts or suggestions feel free and let us know in the comment section.
Additionally, you can read these articles also:
- What is a function? – Definition, Example, and Graph.
- 48 Different Types of Functions and there Examples and Graph – [Complete list].
- What is the Limit of a Function?
- 13 methods to find the Limit of a Function Algebraically