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Understanding polynomial operations is vital for anyone studying algebra. A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables. For instance, in the polynomial given in the exercise, we are working with an expression of the form \(x y^{3}-x y+1\).

When operating with polynomials, it is crucial to follow the standard arithmetic operations and apply them correctly to each term. A term in polynomial is a product of a number, known as the coefficient, and a variable raised to a power, like \(x y^{3}\) or \(-x y\). In this exercise, we are asked to evaluate the polynomial for specific values of \(x\) and \(y\), which will involve substituting these values into the polynomial and simplifying according to the rules of arithmetic operations.

The substitution method, in the context of evaluating polynomials, involves replacing the variables in an expression with specified values and simplifying the result to find a numerical answer. This method plays a crucial role as it turns an algebraic expression into a much simpler arithmetic one. In our example, we substitute \(x\) with 2 and \(y\) with -3, resulting in a new expression that only involves numbers and operations.

Here's a helpful tip to avoid errors: always replace the variables with their values enclosed in parentheses, especially if substituting negative numbers. This helps in maintaining the correct sign and avoiding common mistakes when it comes to dealing with subtraction and negative signs, as seen with \((-3)^{3}\) and \(-2*(-3)\).

The order in which operations should be executed in mathematical expressions is dictated by a set of rules known as the order of operations. Commonly referred to by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), or BODMAS in some regions (Brackets, Orders, Division and Multiplication, Addition and Subtraction), it ensures that calculations are performed in a sequence that gives a correct result.

In the step-by-step solution provided, once the values are substituted, we apply these rules: calculate any expressions within parentheses and exponents first, followed by multiplication or division, and finally, any addition or subtraction. When evaluating the expression \(2*(-27)-2*(-3)+1\), we first deal with multiplication and division, if any, before carrying out the addition and subtraction, which gives us the final answer of -47.