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In algebra, mathematical operations are the building blocks for simplifying expressions and solving equations. They include addition, subtraction, multiplication, division, as well as exponentiation. When working with algebraic expressions like xy(2y-3x), it's crucial to follow the correct order of operations, which is commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

One key aspect of mathematical operations in algebra is combining like terms, which can only be done with terms that have the exact same variables raised to the same powers. For example, xy and 2xy are like terms and can be combined, but xy and yx^2 cannot because the exponents on the x are different.

The distributive property is a cornerstone of algebra that allows us to simplify expressions where a single term is multiplied by a group of terms within parentheses. It states that a(b + c) = ab + ac. This property is essential when working with polynomials and algebraic expressions where multiplication is distributed across addition or subtraction.

For example, in the expression xy(2y-3x), we apply the distributive property to multiply xy with each term inside the parentheses, resulting in x(2y) - x(3x) or, after the multiplication, 2xy - 3x^2. Utilizing this property not only simplifies expressions but also sets the stage for more complex operations in algebra, such as factoring.

Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Examples include 2x + 3, x^2 - 4x + 4, and 3x^3 - 2x^2 + x - 5.

When simplifying expressions with polynomials, it's helpful to identify each term, which is a product of a number (coefficient) and one or more variables raised to integer exponents. In the simplified expression 2xy - 3x^2, for instance, 2xy and -3x^2 are individual terms of a polynomial. Simplifying a polynomial might involve combining like terms, which in this case there are none, and arranging them usually in descending order of their exponents on the variable.