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Zeros Calculator

Find the zeros of a function step by step.

The calculator will try to find the zeros (exact and numerical, real and complex) of the linear, quadratic, cubic, quartic, polynomial, rational, irrational, exponential, logarithmic, trigonometric, hyperbolic, and absolute value function on the given interval.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your input: solve the equation $$$ x^{4} - 16 x^{3} + 90 x^{2} - 224 x + 245=0 $$$ for $$$ x $$$ on the interval $$$ \left( -\infty,\infty \right ) $$$

$$$ x=5 $$$

$$$ x=7 $$$

Complex roots

$$$ x=2 + \sqrt{3} i\approx 2.0 + 1.73205080756888 i $$$

$$$ x=2 - \sqrt{3} i\approx 2.0 - 1.73205080756888 i $$$

Function zeros calculator

One of the task in precalculus is finding zeros of the function - i.e. the intersection points with abscissa axis. Consider the graph of some function :

The zeros of the function are the points at which, as mentioned above, the graph of the function intersects the abscissa axis. To find the zeros of the function it is necessary and sufficient to solve the equation :

The zeros of the function will be the roots of this equation. Thus, the zeros of the function are at the point .

Our online calculator, based on Wolfram Alpha system is able to find zeros of almost any, even very complicated function.

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Zeros Calculator

Zeros Calculator

Write down your function in designated field and the tool will find zeros (real, complex) for it along with their sum and product shown.

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Our real zeros calculator determines the zeros (exact, numerical, real, and complex) of the functions on the given interval.

This tools also computes the linear, quadratic, polynomial, cubic, rational , irrational, quartic, exponential, hyperbolic, logarithmic, trigonometric, hyperbolic, and absolute value function.

What are Zeros of a Function?

In mathematics, the zeros of real numbers, complex numbers, or generally vector functions f are members x of the domain of ‘f’, so that f (x) disappears at x. The function (f) reaches 0 at the point x, or x is the solution of equation f (x) = 0.

Additionally, for a polynomial, there may be some variable values for which the polynomial will be zero. These values ​​are called polynomial zeros. They are sometimes called the roots of polynomials that could easily be determined by using this best find all zeros of the polynomial function calculator . We find the zeros or roots of a quadratic equation to find the solution of a given equation.

Zeros Formula:

Assume that P (x) = 9x + 15 is a linear polynomial with one variable.

Let’s the value of ‘x’ be zero in P (x), then

\( P (x) = 9k + 15 = 0 \)

So, k \( = -15/9 = -5 / 3 \)

Generally, if ‘k’ is zero of the linear polynomial in one variable P(x) = mx + n, then

P(k) = mk + n = 0

k = – n / m

It can be written as,

Zero polynomial K = – (constant / coefficient (x))

How to Find the Zeros of a Function?

Find all real zeros of the function is as simple as isolating ‘x’ on one side of the equation or editing the expression multiple times to find all zeros of the equation. Generally, for a given function f (x), the zero point can be found by setting the function to zero.

The x value that indicates the set of the given equation is the zeros of the function. To find the zero of the function, find the x value where f (x) = 0.

If the degree of the function is \( x^3 + m^{a-4} + x^2 + 1 \), is 10, what does value of ‘a’?

The degree of the function P(m) is the maximum degree of m in P(m).

Therefore, the complex finding zeros calculator takes the \( m^{a-4} = m^4 \)

$$ a-4 = 10, a = 4 + 10 = 14 $$

Hence, the value of ‘a’ is 14.

Calculate the sum and zeros product of the quadratic function \( 4x^2 – 9 \).

The quadratic function is \( 4x^2 – 9 \)

The complex zero calculator can be writing the \( 4x^2 – 9 \) value as \( 2.2x^2-(3.3) \)

Where, it is (2x + 3) (2x-3).

For finding zeros of a function , the real zero calculator set the above expression to 0

$$ (2x + 3) (2x-3) = 0 $$

$$ 2x + 3 = 0 $$

$$ 2x = -3 $$

$$ X = -3/2 $$

Similarly, the zeros of a function calculator takes the second value 2x-3 = 0

$$ 2x = 3 $$

$$ x = 3/2 $$

So, zeros of the function are 3/2 and -3/2

Therefore, zeros finder take the Sum and product of the function:

Zero sum = \( (3/2) + (-3/2) = (3/2) – (3/2) = 0 \)

Zero product = \( (3/2). (-3/2) = -9/4 \).

How this Zeros Calculator Works?

·          Enter an equation for finding zeros of a function .

·          Hit the calculate button to see the results.

·          The real polynomial zeros calculator with steps finds the exact and real values of zeros and provides the sum and product of all roots.

How do you find the roots of a polynomial?

The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x.

Roots Calculator

Find roots of any function step-by-step.

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Frequently Asked Questions (FAQ)

How do you find the root.

What is a root function?

What are complex roots?

How do you find complex roots?


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Zeros Calculator + Online Solver With Free Steps

What is the zeros calculator, what is a zero of a function, what is a real zero, what is the difference between zero and root, features of a zeros calculator, number line representation, sum of roots, product of roots.

The calculated zeros can be real, complex, or exact. The zeros of the real or complex functions are the numerical values at which the function f(x) becomes zero, or in other terms can be written as:

such that x is the zero of the given function in the specified domain.

zeros calculator

A Zeros Calculator is a calculator that can find the zeros of any type of function on any given interval, even the most complicated ones as well.

The Zeros Calculator helps to determine the zeros of the various functions on any given interval. The following is a list of different functions whose zeros can be computed easily and quickly by using this Zeros Calculator:

Hence, the Zeros Calculator helps to solve tedious equations in just seconds. The Zeros Calculator finds the zeros of the given polynomial function with some additional features as well, including the root plot, the sum of the roots, and the product of the roots of the specified function.

How To Use the Zeros Calculator

Let’s discuss how to use the Zeros Calculator to find the zeros of any given function.

The Zeros Calculator helps to find the zeros of any kind of function easily. You can also find the zeros of any function manually, but it requires a lot of time and is a very lengthy procedure in terms of numerical calculations.

Methods to find Zeros

Figure-1 Methods to find Zeros

Therefore, with the help of this calculator, you can step toward your desired results smartly and save much more time. You just have to follow these simple steps to find the zeros of any function.

Use the Zero Calculator to find the zeros of the desired function.

There is an expression tab in the calculator. Input the function here for which the zeros are required to be calculated.

After you have entered the function for which you want to find the zeros, press the submit button placed just below the expression tab.

Once you have pressed the submit button, a new window will appear in front of you displaying the results. Zeros Calculator finds the zeros of the given function along with a root plot, zeros represented on a number line, sum of zeros, and product of zeros.

Lastly, for the detailed and step-by-step solution, you just have to click on the appropriate button given for the detailed solution and you can view the steps. If you want to find the roots of any other function, enter the new equation in the expression tab and follow the same procedure as mentioned above.

How Does a Zero Calculator Work?

A Zeros Calculator works by setting the function equivalent to zero and calculating the zeros. It works by segregating the variable x on one side of the equation or modifying the specified equation several times to find out all the zeros of the function. Let’s have a deep insight into the concept of function zeros.

Finding the roots or zeros of any type of function manually is very cumbersome and error-prone. There can be a polynomial with lots of roots that can be nearly impossible for you to calculate by hand, but this online zeros calculator has got you covered. You can calculate the zeros quickly by just simply entering the desired function into it.

The zero of the function is the point that corresponds to the values of the variable of a function that when put in the function, the function becomes zero. Graphically, zero of the function is the point where it intersects the x-axis. In other terms, it can also be called x-intercepts of the graph of the function.

To find the value of the zero for the given function, set the function equal to zero and then calculate the value of the variable of the function; the corresponding values are called Zeros. To further simplify the concept, Zero of the function is defined as the point where the function becomes zero or crosses the x-axis of the graph of a function.

Another important thing to consider is that a function can have more than one zero depending upon the degree of the polynomial or function. A degree of function is defined as the highest degree of its variable. Therefore, the total number of zeros of any function depends upon the degree of the function. 

For instance, to further clarify this concept, a Linear function is a degree 1 function. Hence, all the linear functions have only one zero. Similarly, a Quadratic function is a second-degree function, therefore all the quadratic functions have two zeros or it intersects the x-axis of the graph of a function at two points.

A zero is said to be a Real zero if it belongs to the set of a real number provided that the function of value becomes zero. If f(x) = 0 where x $\in$ $\mathbb{R}$, then x is called a real zero of the function.

The main difference between zero and root is that zero is associated with a function whereas a root refers to an equation. A zero of a function is a value at which the function becomes zero as x is referred to as a root of the function f(x) if and only if the f(x) becomes equal to zero.

A root of an equation is the value of its variable x at which the equation is satisfied or both sides of the equation become equal. A polynomial equation can also have more than one root depending upon the degree of the polynomial equation.

A Zeros Calculator is a very useful tool as it not only provides you with the roots of the function, but it also has some additional features listed below:

Features of Zeros

Figure-2 Features of Zeros

A root plot is a graphical representation of all the roots of the function. It shows the graph of a function with the indication of x-intercepts that are the zeros of the function.

The zeros calculator also represents the zeros of the function on the number line. A number line is defined as the line on which various points are marked at various intervals.

The zeros calculator also provides the sum of all the roots of the function.

Lastly, it also calculates the product of all the roots of the function.

Solved Examples

Find the roots of the given function using the Zeros Calculator. Draw the root plot and number line representation of the zeros. Also, find the sum and product of the roots of the function.

\[ f(x) = x^2-8 \]

Enter the given function in the expression tab of the Zeros Calculator.

It will display the following results:

The roots of the function are given as:

\[ x = + 2 \sqrt{2} \]

\[ x = – 2 \sqrt{2} \]

The root plot is shown in Figure 1:

Zeros of function three

Figure 3 Zeros of function three

Zeros represented on Number Line are shown in Figure 2:

Zeros of function four

Figure 4 Zeros of function four

The sum of all the roots:

product = – 8 

Find the zeros of the following trigonometric function: 

\[  f(x) = 2 sin x + \sqrt{3} \]

Use the calculator to find the roots.

Enter the given function in the expression tab of the Zeros Calculator to find the zeros of the function.

\[ x = \dfrac{2}{3} \pi ( 3n + 2) \]

\[ x = \dfrac{1}{3} \pi ( 6n – 1)  \]

Find the zeros of the following function given as:

\[  f(x) = x^4 – 16 \]

This polynomial function has 4 roots (zeros) as it is a 4-degree function. It has two real roots and two complex roots

It will display the results in a new window.

x = – 2 

x = – 2i

Find the zeros of the following polynomial  function: 

\[  f(x) = x^4 – 4x^2 + 8x + 35 \]

This is a polynomial function of degree 4. Therefore, it has four roots.

All the roots lie in the complex plane.

x = -2 – i

\[ x = 2 – \iota \sqrt{3} \]

\[ x = 2 + \iota\ \sqrt{3} \]

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How to find polynomial roots ?

The process of finding polynomial roots depends on its degree . The degree is the largest exponent in the polynomial. For example, the degree of polynomial $ p(x) = 8x^\color{red}{2} + 3x -1 $ is $\color{red}{2}$.

We name polynomials according to their degree. For us, the most interesting ones are: quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4.

Roots of quadratic polynomial

This is the standard form of a quadratic equation

The formula for the roots is

Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $

In this case we have $ a = 2, b = 3 , c = -14 $, so the roots are:

Quadratic equation - special cases

Sometimes, it is much easier not to use a formula for finding the roots of a quadratic equation.

Example 02: Solve the equation $ 2x^2 + 3x = 0 $

Because our equation now only has two terms, we can apply factoring . Using factoring we can reduce an original equation to two simple equations.

Example 03: Solve equation $ 2x^2 - 10 = 0 $

This is also a quadratic equation that can be solved without using a quadratic formula.

The last equation actually has two solutions. The first one is obvious

and the second one is

Roots of cubic polynomial

To solve a cubic equation, the best strategy is to guess one of three roots.

Example 04: Solve the equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $.

Step 1: Guess one root.

The good candidates for solutions are factors of the last coefficient in the equation. In this example, the last number is -6 so our guesses are

1, 2, 3, 6, -1, -2, -3 and -6

if we plug in $ \color{blue}{x = 2} $ into the equation we get,

So, $ \color{blue}{x = 2} $ is the root of the equation. Now we have to divide polynomial with $ \color{red}{x - \text{ROOT}} $

In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$.

Now we use $ 2x^2 - 3 $ to find remaining roots

Cubic polynomial - factoring method

To solve cubic equations, we usually use the factoting method:

Example 05: Solve equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $.

Notice that a cubic polynomial has four terms, and the most common factoring method for such polynomials is factoring by grouping.

Now we can split our equation into two, which are much easier to solve. The first one is $ x - 2 = 0 $ with a solution $ x = 2 $, and the second one is $ 2x^2 - 3 = 0 $.

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Polynomial Zeros

Instructions: Use calculator to find the polynomial zeros, showing all the steps of the process, of any polynomial you provide in the form box below.

algebra zeros calculator

This calculator will allow you compute polynomial roots of any valid polynomial you provide. This polynomial can be any polynomial of degree 1 or higher.

For example, you can provide a cubic polynomial, such as p(x) = x^3 + 2x^2 - x + 1, or you can provide a polynomial with non-integer coefficients, such as p(x) = x^3 - 13/12 x^2 + 3/8 x - 1/24.

Once you have provided the calculator with a valid polynomial for which you want to compute its roots, you can click on the "Calculate" button, and you will see a step-by-step run of the process.

It needs be mentioned that the process only involve elementary methods used to find roots, which includes the rational zero theorem and polynomial division , as well as using the quadratic formula when appropriate.

There is no general method to find ALL the roots for ALL possible polynomials of degree above 5, so this calculator will only find roots that can be obtained with these mentioned elementary methods.

Polynomial Zeros

What is the root of a polynomial?

Given a polynomial function \(p(x)\), we say that \(x\) is a root of the polynomial if:

In layman terms roots of a polynomial are the points the the polynomial function \(p(x)\) crosses the x-axis. That's a good representation to get an idea, but it is not completely precise because some roots could be complex numbers. So then, a real root will be a point where \(p(x)\).

Observe that the roots of the polynomial are also called polynomial zeros.

What are the steps for finding the zeros of a polynomial?

It is usually not easy, and it can be computationally intensive, and it is not guaranteed to work, but it is the best possible approach if we are restricted to using elementary methods.

Is factoring the only way to find roots

Not really, but things go hand-to-hand. The factor theorem states that \(x - a\) is a factor of a polynomial \(p(x)\) if and only if \(p(a) = 0\). So in other words, roots and factors are intimately linked.

Now, for polynomials of degree 2 (this is, quadratic polynomials ) we can use an explicit formula, which is the well know quadratic formula .

The same happens for degrees 3 and 4, though the formulas are far from elementary. But for degree 5 and higher, there is no such formula, a key result proven by Galois and Abel. So there is no hope to find a "general formula", and which is why the use a more lax polynomial factorization approach.

Common mistakes to avoid

Often times students get frustrated that they cannot find the roots of a given polynomial function , say \(p(x) = x^3+2 x^2-x+1 \), but they need to face the fact that not all polynomials will be able to be solved using elementary tools.

Granted, there is a formula to solve \(x^3+2 x^2-x+1 = 0 \), but it is not elementary, and it is not expected that students know it.

Tips for success

Always try to do a mental map of what your strategy will be: Take note of the polynomial you have, its degree, its leading coefficient and constant coefficient.

Plot the polynomial if you can, to get an idea of its behavior. Are there any obvious factorization that you can use? Use them. Always remember factors = roots.

Polynomial Roots

Example: Zeros of a Polynomial

What are the zeros of : \(x^5 + x^4 - x^3 + x^2 - x + 1\)?

Solution: For this example we are provided with the following polynomial: \(\displaystyle p(x) = x^5+x^4-x^3+x^2-x+1\). We will use the factoring approach to finding roots.

Simplification not needed: The provided polynomial expression is simplified already, so there is nothing to simplify it further.

It can be noted that the degree of the provided polynomial is \(\displaystyle deg(p) = 5\). Also, its leading coefficient is \(\displaystyle a_{5} = 1\) and its constant coefficient is equal to\(\displaystyle a_0 = 1\).

Now we search for integer numbers that divide the leading coefficient \(a_{5}\) and the constant coefficient \(a_0\), which is used to find rational candidates .

▹ The dividers of \(a_{5} = 1\) are: \(\pm 1\).

▹ The dividers of \(a_0 = 1\) are: \(\pm 1\).

Therefore, dividing all factors of the constant term \(a_0 = 1\) by all the dividers of \(a_{5} = 1\), we get the following list of potential roots:

Now, all potential solutions must be evaluated. The results obtained from testing each candidate are as follows:

Since no rational roots were identified through manual inspection, further simplification using basic techniques is not possible and the process ends with this step.

Conclusion : As a result, no simplification was obtained and no roots of the polynomial were identified through basic techniques

Example: Calculating roots a quadratic function

Calculate the solutions of: \(3x^2 - 2x - 4 = 0\).

Solution: We need to solve the given quadratic equation \(\displaystyle 3x^2-2x-4=0\).

The roots for a quadratic equation of the form \(a x^2 + bx + c = 0\) are calculated using the following equation:

In this context, the equation that needs to be solved is \(\displaystyle 3x^2-2x-4 = 0\), indicating that the corresponding coefficients are:

First, we will determine the nature of the roots by calculating the discriminant. The discriminant is calculated as follows:

Since in this case we get the discriminant is \(\Delta = \displaystyle 52 > 0\), which is positive, so then, the equation has two different real roots.

From this we get:

so then, we find that:

We find that the equation \( \displaystyle 3x^2-2x-4 = 0 \), has two real roots, so then:

so then the original polynomial is factored as \(\displaystyle p(x) = 3x^2-2x-4 = 3 \left(x+\frac{1}{3}\sqrt{13}-\frac{1}{3}\right)\left(x-\frac{1}{3}\sqrt{13}-\frac{1}{3}\right) \), which completes the factorization.

Conclusion : Therefore, the factorization we get looking for is given by:

The roots found are \(-\frac{1}{3}\sqrt{13}+\frac{1}{3}\) and \(\frac{1}{3}\sqrt{13}+\frac{1}{3}\) .

Example: Polynomial Zeros

Calculate the zeros of the following polynomial: \(p(x)= x^3 - \frac{13}{12} x^2 + \frac{3}{8} x - \frac{1}{24} \).

Solution: Finally, in this example we have: \(\displaystyle p(x) = x^3-\frac{13}{12}x^2+\frac{3}{8}x-\frac{1}{24}\).

First Step: The provided polynomial expression is irreducible, so there is nothing to simplify. We can proceed to factor it.

Observe that the degree of the given polynomial is \(\displaystyle deg(p) = 3\), its leading coefficient is \(\displaystyle a_{3} = 1\) and its constant coefficient is \(\displaystyle a_0 = -\frac{1}{24}\).

Rational Roots : We will try to find simple rational roots first, with the Rational Zero Theorem.

The next task is to find the integer numbers that divide the leading coefficient \(a_{3}\) and the constant coefficient \(a_0\), that will be used to construct our candidates to be zeroes of the polynomial equation.

Note: In this case, we observe that in order to have both constant and leading coefficient we need to amplify both sides of the equation by \(24\). The equivalent equation is:

▹ The dividers of \(a_{3} = 24\) are: \(\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12,\pm 24\).

▹ The dividers of \(a_0 = -1\) are: \(\pm 1\).

Therefore, dividing each divider of the constant coefficient \(a_0 = -1\) by each divider of the leading coefficient \(a_{3} = 24\), we find the following list of candidates to be roots:

Now, all the candidates need to be tested to see if they are a solution. The following is obtained from testing each candidates:

But since we have found all the required roots among the rational candidates, we find that \(\displaystyle x^3-\frac{13}{12}x^2+\frac{3}{8}x-\frac{1}{24} = \left(x-\frac{1}{2}\right)\left(x-\frac{1}{3}\right)\left(x-\frac{1}{4}\right) \), so then:

which completes factorization process.

Result : Therefore, the final factorization is:

Therefore, the roots found are \(\frac{1}{2}\),\(\frac{1}{3}\) and \(\frac{1}{4}\) .

Other useful polynomial calculators

Finding zeros of a polynomial is one the pinnacles of Algebra, to the degree that the Fundamental Theorem of Algebra is about the existence of n roots for a polynomial of degree n. Those roots will not necessary be all real, and some of them (or all of them) may be compex numbers.

Ultimately, almost every single problem in Algebra and Calculus can be reduced to finding roots of a polynomial, including solving polynomial equations , such as the ones you would find for example, when looking for the intersection between the graphs of \(y = x^2\) and \(y = x^3\).

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