How can we find the root for high degree polynomial functions?

[calculateme]

How do you find the root for high degree polynomial functions? I am really lost on this, any help would be appreciated.

 

[NateTG]

I assume that you're referring to real polynomials.

For polymials of degree 5 or greater there is no general algebraic solution.

There are excellent numerical methods for polynomials of finite order, but to understand how they work, requires a little calculus.

Algebraically, there is a field of mathematics called Galois theory that deals with finding roots of polynomials.

 

[tacman]

The root of a polynomial function is any value for the independent variable (usually denoted as x) that makes the function equal to zero. In other words, it is the value of x that "cancels out" the polynomial and yields a result of 0.

To find the root of a polynomial function, we can use a method called the "zero product property." This method states that if a product of two or more factors equals zero, then at least one of the factors must be equal to zero. In the case of a polynomial function, the factors are the individual terms of the polynomial.

For example, let's say we have the polynomial function f(x) = x^2 + 3x - 10. To find the roots of this function, we can set the function equal to zero and solve for x:

x^2 + 3x - 10 = 0

Using factoring or the quadratic formula, we can find that the roots of this function are x = 2 and x = -5.

For high degree polynomial functions (functions with a degree greater than 2), finding the roots can be more challenging. In these cases, it may be helpful to use a graphing calculator or software to plot the function and visually determine the roots. Another method is to use the Rational Root Theorem, which provides a list of potential rational roots based on the leading coefficient and constant term of the polynomial.

In addition, if the polynomial function can be factored, we can use the zero product property to find the roots. However, not all polynomial functions can be factored, especially those with a high degree.

In summary, finding the roots of polynomial functions can be done through various methods such as factoring, using the zero product property, or using the Rational Root Theorem. It is also helpful to use technology or graphing to assist in finding the roots for high degree polynomial functions.