Finding zeros of a function is one of the most fundamental skills in algebra, calculus, and beyond. Whether you’re solving a quadratic equation, analyzing a graph, or working with polynomials, the ability to locate where a function equals zero is essential. Zeros, also known as roots or x-intercepts, represent the points where a function crosses or touches the x-axis. Understanding how to find zeros allows students and professionals to interpret mathematical behavior, solve real-world problems, and simplify complex expressions.
What Are Zeros of a Function?
Zeros are the values of x that make a function f(x) equal to zero. In other words, they are the solutions to the equation f(x) = 0. These points can be found algebraically or visually by examining a graph.
Examples of Zeros
- If f(x) = x – 5, then the zero is x = 5 because f(5) = 0.
- If f(x) = x² – 9, the zeros are x = 3 and x = -3 since f(3) = 0 and f(-3) = 0.
Zeros can be real or complex, depending on the type of equation and its degree. Finding them accurately is an important skill in solving mathematical problems and analyzing data trends.
Basic Techniques to Find Zeros
There are multiple methods to find the zeros of a function, depending on its form. The most common techniques are:
1. Factoring
Factoring is often the easiest method for finding zeros when dealing with polynomials, especially quadratics or cubics with rational roots.
Example:
Let f(x) = x² – 4x + 3.
Factor the expression: f(x) = (x – 1)(x – 3).
Set each factor to zero: x – 1 = 0 and x – 3 = 0.
Solutions: x = 1 and x = 3.
2. Using the Quadratic Formula
When a quadratic function cannot be factored easily, the quadratic formula is a reliable alternative:
x = (-b ± â(b² – 4ac)) / 2a
This formula works for any equation in the form ax² + bx + c = 0.
Example:
f(x) = 2x² + 3x – 2
a = 2, b = 3, c = -2
Apply the formula: x = (-3 ± â(3² – 4à 2à -2)) / (2à 2)
Find the values of x, which are the zeros of the function.
3. Graphing
Another way to find zeros is to graph the function and identify the x-intercepts visually. This method is useful when approximate values are acceptable or when using graphing technology like a calculator or software.
Steps:
- Plot the function on a graph.
- Locate the points where the curve intersects the x-axis.
- The x-coordinates of those points are the zeros of the function.
4. Synthetic Division and Rational Root Theorem
For higher-degree polynomials, the Rational Root Theorem helps list possible rational zeros. Then, synthetic division can be used to test them and reduce the polynomial’s degree.
Example:
f(x) = 2x³ – 3x² – 2x + 3
List possible rational roots: ±1, ±3, ±1/2, ±3/2
Use synthetic division to test each value.
Once a root is found, divide the polynomial to get a simpler expression and repeat the process.
Special Cases in Finding Zeros
Repeated Zeros (Multiplicity)
Sometimes a function has a zero that appears more than once. This is known as a multiple root or having multiplicity greater than one.
Example:
f(x) = (x – 2)²(x + 1)
The zeros are x = 2 (with multiplicity 2) and x = -1.
At x = 2, the graph touches the x-axis but does not cross it.
Complex Zeros
When the discriminant (b² – 4ac) is negative in the quadratic formula, the function has complex zeros.
Example:
f(x) = x² + 4
Using the formula: x = (0 ± â(-16)) / 2
Zeros are x = 2i and x = -2i, which are complex numbers.
Zeros of Rational Functions
To find the zeros of a rational function, set the numerator equal to zero and solve. Do not include values that make the denominator zero, as they are undefined.
Example:
f(x) = (x – 3)/(x² – 4)
Set numerator to zero: x – 3 = 0 â x = 3 is the zero.
The denominator x² – 4 = 0 gives x = ±2, which are excluded from the domain.
Checking and Verifying Zeros
After finding the zeros, it’s important to verify them by plugging the values back into the original equation. If f(x) equals zero at those points, the solution is correct.
Example:
f(x) = x² – 4x + 3
Found zeros: x = 1, x = 3
Check: f(1) = 1 – 4 + 3 = 0 â
f(3) = 9 – 12 + 3 = 0 â
Applications of Zeros in Real Life
Finding zeros is not just a theoretical exercise. It has practical uses in fields like physics, engineering, economics, and data science.
Examples of Applications
- In physics, zeros indicate the points where velocity is zero or an object changes direction.
- In business, zeros help determine breakeven points where profit is zero.
- In engineering, zeros are essential in signal processing and control systems.
Using Technology to Find Zeros
Many modern tools can help find zeros quickly and accurately. Graphing calculators, online solvers, and computer software like Desmos, WolframAlpha, or GeoGebra are useful when dealing with complex equations or checking manual work.
Advantages of Technology
- Saves time for long or complicated expressions
- Visual representation helps understand the function’s behavior
- Accuracy in solving decimals and irrational roots
Knowing how to find zeros of a function is a crucial mathematical skill that serves as the foundation for solving equations, analyzing functions, and applying math in practical situations. Whether using factoring, the quadratic formula, graphing, or synthetic division, the goal is to identify the values of x that make the function equal zero. With consistent practice and understanding of the different methods, students can master the process and apply it confidently in various areas of math and science.