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Root finding algorithms are essential in numerical analysis and applied mathematics for determining the zeroes of a function. A 'zero' or 'root' of a function is a point where the function's value is zero. When dealing with polynomials, finding these roots can be quite challenging, especially as the degree of the polynomial increases.

There are various algorithms available to find roots, from simple bisection methods to more complex algorithms like Newton-Raphson and Bairstow's method. The choice of algorithm depends on the type of function, the desired accuracy, and computational efficiency. Bairstow's method, for instance, is particularly suited for polynomials and operates iteratively, refining guesses of the roots until a satisfactory approximation is reached.

Dealing with polynomial equations, numerical methods are often the only way to find roots, particularly when exact algebraic solutions are not feasible. Bairstow's method is one of these numerical methods applied specifically to polynomials. It is an alternative to other methods like the Durand-Kerner method or the Jenkins-Traub algorithm.

Polynomial equations can have multiple roots, including real and complex ones. Numerical methods iteratively approach these roots by making initial guesses and then refining these guesses using a predetermined formula or set of rules that guide the estimation towards the actual root.

Synthetic division is a simplified method of dividing a polynomial by a divisor of the form \(x - k\) without having to write out all the terms and their coefficients. It's a key component of Bairstow's method. When using this method to assist in finding roots, you set up a scheme that allows the division to be carried out quickly and efficiently, using only the coefficients of the polynomial.

Following this scheme, iterating over the coefficients of the polynomial, one can obtain the remainder after 'dividing' by an assumed quadratic factor. In Bairstow's method, synthetic division is used iteratively to improve the guessed quadratic factors until the remainder is minimized.

Iterative methods are central in numerical analysis for solving various mathematical problems. These methods start with an initial guess and then utilize a sequence of operations to approach the desired solution. The process is repeated, i.e., iterated, until the solution is within an acceptable range of error.

Iterative methods, such as Bairstow's method, are designed to improve the accuracy of the solution with each iteration, based on feedback from the previous step's result. This allows for a dynamic approach to solving complex equations which may not have straightforward analytical solutions. Iterative methods are favoured for their flexibility and adaptability in handling a wide range of problems.