Finding all zeros of a polynomial calculator

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.

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

\( a \) 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.)

More References and Links

polynomials
Factor Polynomials
Find Zeros of Polynomials
Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8

x^2x^{\msquare}\log_{\msquare}\sqrt{\square}\nthroot[\msquare]{\square}\le\ge\frac{\msquare}{\msquare}\cdot\divx^{\circ}\pi\left(\square\right)^{'}\frac{d}{dx}\frac{\partial}{\partial x}\int\int_{\msquare}^{\msquare}\lim\sum\infty\theta(f\:\circ\:g)H_{2}O

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.