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When simplifying algebraic expressions, especially those involving subtraction, one key step is distributing the negative sign. This involves changing the signs of terms inside parentheses when preceded by a subtraction. Consider the expression: \[x^2y + 3xy^2 - (2x^2y - xy^2)\] Here, the negative sign before the parentheses affects each term inside. So:- Every term inside becomes its opposite due to this subtraction.- \(-2x^2y\) turns into \(-2x^2y\) (no change in prerequisite sign). - However, \(-xy^2\) becomes \(+xy^2\). After distributing the negative sign, the expression transforms to: \[x^2y + 3xy^2 - 2x^2y + xy^2\]This critical step sets you up for the next process of combining like terms.

After distributing the negative sign and simplifying your expression, the next step is to combine like terms. Like terms are those with the same variables raised to the same powers.Given the expression: \[(x^2y + 3xy^2 - 2x^2y + xy^2)\]Identify like terms:- \(x^2y\) and \(-2x^2y\) are like terms. They share the same variables and exponents.- \(3xy^2\) and \(xy^2\) are another pair of like terms.To combine them:- Subtract coefficients for \(x^2y\) terms: \[x^2y - 2x^2y = -x^2y\]- Add coefficients for \(xy^2\) terms: \[3xy^2 + xy^2 = 4xy^2\]Always pay attention to the signs of the coefficients when doing this step, as it impacts the final result.

Polynomial expressions are mathematical phrases combining variables, coefficients, and exponents, employing the operations of addition, subtraction, and multiplication. In this context, the expression \[-x^2y + 4xy^2\]is a simplified version of a polynomial expression.Understanding polynomials:- **Terms**: Each part of a polynomial separated by a plus (+) or minus (-) sign. In our case, \(-x^2y\) and \(4xy^2\) are distinct terms.- **Degree**: The highest sum of exponents of variables in a single term. Here, \(-x^2y\) has the highest degree of 3 (since \(x^2y = x^2y^1\)).Polynomials can be simple or complicated, but the principles of distributing negative signs and combining like terms help in simplifying them effectively. Once simplified, this structure allows for easier computation and understanding of algebraic relationships.