Find all complex zeros of the polynomial function calculator

Q: How do you find all real zeros of a polynomial function?

To find the real zeros of a polynomial, first convert the polynomial to factored form. Once all factors are found, set each individual factor equal to zero to solve for the real zeros.

Q: Do all polynomials have complex zeros?

Every polynomial function of positive degree n has exactly n complex zeros (counting multiplicities). For example, P(x) = x5 + x3 - 1 is a 5th degree polynomial function, so P(x) has exactly 5 complex zeros.

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Table of Contents Show

Zeros of a Polynomial
Use of the zeros Calculator
More References and Links
Online Equation Solver
Solve linear, quadratic and polynomial systems of equations with Wolfram|Alpha
More than just an online equation solver
Tips for entering queries
Access instant learning tools
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About solving equations
A value is said to be a root of a polynomial if .
How Wolfram|Alpha solves equations
How do you find all real zeros of a polynomial function?
Do all polynomials have complex zeros?

A calculator to calculate the real and complex zeros of a polynomial is presented.

Zeros of a Polynomial

\( a \) is a zero of a polynomial \( P(x) \) if and only if \( P(a) = 0 \)
or
\( a \) is a zero of a polynomial \( P(x) \) if and only if \( x - a \) is a factor of \( P(x) \)
Note that the zeros of the polynomial \( P(x) \) refer to the values of \( x \) that makes \( P(x) \) equal to zero. But both the zeros and the roots of a polynomial are found using factoring and the factor theorem [1 2].

Example
Find the zeros of the polynomial \( P(x) = x^2 + 5x - 14 \).
Solution
Factor \( P(x) \) as follows
\( P(x) = (x-2)(x+7) \)
Set \( P(x) = 0 \) and solve
\( P(x) = (x-2)(x+7) = 0 \)
Apply the factor theorem [1 2] and write that each factor is equal to zero.
\( x-2 = 0 \) or \( x+7 = 0 \)
Solve to obtain
\( x = 2 \) and \( x = - 7 \)
Hence the zeros of \( P(x) \) are \( x = 2 \) and \( x = - 7 \)

Use of the zeros Calculator

1 - Enter and edit polynomial \( P(x) \) and click "Enter Polynomial" then check what you have entered and edit if needed.
Note that the five operators used are: + (plus) , - (minus), , ^ (power) and * (multiplication). (example: P(x) = -2*x^4+8*x^3+14*x^2-44*x-48).(more notes on editing functions are located below)
2 - Click "Calculate Zeros" to obain the zeros of the polynomial.
Note that the zeros of some polynomials take a large amount of time to be computated and their expressions may be quite complicated to understand.

Notes: In editing functions, use the following:
1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: P(x) = 2*x^2 - 2*x - 4 )
Here are some examples of polynomials that you may copy and paste to practice:
x^2 - 9 x^2 + 9 x^2 + 2*x + 7 x^3 + 2*x - 3 3*x^4 - 3
x^5+5*x^4+3*x^3+x^2-10*x-120 x^5+4x^4-7x^3-28x^2+6x+24
x^4 - 4*x^3 + 3 (this one has very complicated zeros and takes time to compute; try it to have an idea.)

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Wolfram|Alpha is a great tool for finding polynomial roots and solving systems of equations. It also factors polynomials, plots polynomial solution sets and inequalities and more.

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Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to formulate queries.

find roots to quadratic x^2-7x+12
plot inequality x^2-7x+12<=0
solve {3x-5y==2,x+2y==-1}

plot inequality 3x-5y>=2 and x+2y<=-1
solve 3x^2-y^2==2 and x+2y^2==5
plot 3x^2-y^2>=2 and x+2y^2<=5

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About solving equations

A value is said to be a root of a polynomial if .

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

How Wolfram|Alpha solves equations

For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. Other operations rely on theorems and algorithms from number theory, abstract algebra and other advanced fields to compute results. These methods are carefully designed and chosen to enable Wolfram|Alpha to solve the greatest variety of problems while also minimizing computation time.

Although such methods are useful for direct solutions, it is also important for the system to understand how a human would solve the same problem. As a result, Wolfram|Alpha also has separate algorithms to show algebraic operations step by step using classic techniques that are easy for humans to recognize and follow. This includes elimination, substitution, the quadratic formula, Cramer's rule and many more.

How do you find all real zeros of a polynomial function?

To find the real zeros of a polynomial, first convert the polynomial to factored form. Once all factors are found, set each individual factor equal to zero to solve for the real zeros.

Do all polynomials have complex zeros?

Every polynomial function of positive degree n has exactly n complex zeros (counting multiplicities). For example, P(x) = x⁵ + x³ - 1 is a 5^th degree polynomial function, so P(x) has exactly 5 complex zeros.