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Understanding the concept of polynomial roots is fundamental to solving equations in algebra. A polynomial root, also known as a zero, is a value for which the polynomial equals zero. In simple terms, when you substitute this value into the polynomial, the output is zero. Finding roots of a polynomial is like solving a puzzle—each root provides a piece of the solution.

For example, consider the polynomial equation \(x^{4}-4x^{3}-6x^{2}+16x-4=0\). The roots are the values of x that make this equation true. These roots can be real numbers or complex numbers (which include an imaginary part). Complex roots for polynomials of degree four or higher are common, and they always come in conjugate pairs, meaning if \(a + bi\) is a root, then \(a - bi\) will be too, where \(i\) is the imaginary unit.

It's crucial to keep in mind that a polynomial of degree \(n\) will have \(n\) roots, if you consider the multiplicity and complex roots as well. The roots are not always rational or even real numbers; they can be irrational or complex, which can make finding them a more challenging task. This leads to other methods such as synthetic division or numerical algorithms to help in determining these values.

When it comes to breaking down polynomials, synthetic division is a handy tool, especially if you're working with higher degree polynomials. It's a simplified method of dividing a polynomial by a linear factor and can be particularly useful when finding possible rational roots.

Synthetic division streamlines the process of long division into a series of simpler arithmetic operations. It involves lining up coefficients and performing multiplication and addition operations. The result tells us not only the quotient but also the remainder of the division. If the remainder is zero, we have found a root!

However, when possible rational roots, such as \(\text{1, -1, 2, -2, 4, -4}\) here, do not actually satisfy the polynomial equation, synthetic division alone won't help us find the solution. This may prompt the need for numerical methods to approximate the roots, particularly when dealing with complex or irrational solutions that can't be easily factored.

When other methods fail to find the roots of a polynomial equation, we turn to numerical methods for root finding. These methods provide ways to approximate the polynomial's roots, especially when they are not rational or easily factored.

Algorithms like Newton-Raphson, Bisection, and Secant are commonly used for this purpose. Newton-Raphson, for instance, is an iterative method which starts with a guess and iteratively improves the estimate of the root. It's a powerful technique but also requires knowledge of calculus, as it involves the derivative of the function. The Bisection method repeatedly narrows down an interval containing the root by halving it and selecting the subinterval where the function changes sign.

These are just a few options among many other sophisticated numerical techniques. In some cases, specially designed software and graphing calculators apply these methods internally to rapidly compute the roots to a high degree of accuracy. While these methods can be complex, they are incredibly valuable for solving equations that are impossible to factor or solve analytically, which is often the case in the advanced study of mathematics and its applications in various fields.