# Polynomial factorization calculator and root finder

#### alpertron2

Hello folks.

I have written a Web calculator that can factor polynomials and it can also find exact roots using rational numbers and radical expressions.

You can enter any polynomial expression in the input box.

You can see it at:

http://www.alpertron.com.ar/POLFACT.HTM

Currently, it can find exact roots of polynomials with degrees up to 4. For degree 5, it can say whether the polynomial can be solved or not. In the first case, it shows the solutions in most cases (except when the polynomial has 5 real roots, where trigonometric solutions are required).

Let me know whether it is useful for you, whether more feature can be added, etc.

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#### skipjack

Forum Staff
You should use the word "zeros" instead of "roots".

Did you mean "in the second case"? In the first case (polynomials of degrees up to 4), 5 real zeros would be impossible.

#### alpertron2

You should use the word "zeros" instead of "roots".
According to http://www.mathopenref.com/rootpolynomial.html, The roots of a polynomial are those values of the variable that cause the polynomial to evaluate to zero.
Did you mean "in the second case"? In the first case (polynomials of degrees up to 4), 5 real zeros would be impossible.
In order to obtain the roots of an irreducible fifth degree polynomial, you have to compute a sixth degree resolvent.

There are two cases:

In the first case, the resolvent has a rational root. Using this root and solving other polynomials you can find exact expressions for the five roots.

In the second case the resolvent does not have a rational root. In this case the roots cannot be expressed by radical expressions.

This means that most fifth degree polynomials cannot be expressed by radical expressions.

In some cases (when all five roots are real), you will have to compute fifth roots of complex numbers. You can compute the roots using real numbers by using trigonometric functions. This is the remaining part to be done in the application.

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#### skipjack

Forum Staff
According to this webpage, the word "root" means "a solution to an equation of the form f(x) = 0. The roots of f(x) = 0 are the same as the zeros of the function f(x). Sometimes in casual usage the words root and zero are used interchangeably." I prefer the stated meanings to the casual usage mentioned, even if the "casual" usage has become common. I suspect that using "root" as a synonym for "zero" is less common in mathematical journals and on university web sites.

Your explanation of two cases hadn't been given prior to your earlier reference to "the first case", making your earlier post difficult to understand. Your sentence structure doesn't require that "the first case" is referring to the immediately preceding sentence, as distinct from the sentence before that.